W-supplemented modules

GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

W-SUPPLEMENTED MODULES

by

Tu˘gba G ¨

URO ˘

GLU

July, 2010 ˙IZM˙IR

A Thesis Submitted to the

Graduate School of Natural and Applied Sciences of Dokuz Eyl ¨ul University In Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in

Mathematics

by

Tu˘gba G ¨

URO ˘

GLU

July, 2010 ˙IZM˙IR

We have read the thesis entitled “ W-SUPPLEMENTED MODULES ” completed by TU ˘GBA G ¨URO ˘GLUunder supervision of ASSOC. PROF. G ¨OKHAN B˙ILHAN and we certify that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.

... 11111111111111111111111111Assoc. Prof. G¨okhan B˙ILHAN

Supervisor

... 11111111111111111111111111 Assistant Prof. Engin MERMUT

Thesis Committee Member

... 11111111111111111111111111

Assistant Prof. Hakan EP˙IK

Thesis Committee Member

... 11111111111111111111111111

Examining Committee Member

... 11111111111111111111111111

Examining Committee Member

11111111111111111111111111 Prof. Dr. Mustafa SABUNCU

Director

Graduate School of Natural and Applied Sciences

I would like to express my deepest gratitude to my supervisor Assoc. Prof. G¨okhan B˙ILHAN for his advice, guidance, encouragement, never ending patience and infinite support during preparation of this thesis and I wish to thank all staff Dokuz Eyl¨ul University Department of Mathematics for their contribution to my education. I want to thank Engin MERMUT for his suggestions. Also, I would like to thank Christian LOMP for his help in my work.

Finally, I am very grateful to my family, especially my mother Hatice G ¨URO ˘GLU, for their support, understanding and encouragement in my life.

Tu˘gba G¨uro˘glu

ABSTRACT

The purpose of this thesis is to investigate the properties of w-supplemented, totally

w-supplemented and w-coatomic modules. A module M is w-supplemented module if

and only if M is amply w-supplemented module. Although the class of w-supplemented modules is closed under extension with certain condition, the class of w-coatomic modules is closed under extension by short exact sequence. We give an example of a module M for which every submodule of w-supplemented module need not be

w-supplemented. Every torsion module is (totally) w-supplemented over a Dedekind

domain. If R is a semilocal ring, then every R-module is (totally) w-supplemented module. We prove that a module M is w-supplemented if and only if M is totally

w-supplemented over a left V -ring. If M is a reduced and w-supplemented module,

then M is w-coatomic.

Keywords: Semisimple, radical, socle, w-supplemented, totally w-supplemented,

w-coatomic.

¨ OZ

Bu tezde w-t¨umlenmis¸, t¨umden w-t¨umlenmis¸ ve w-koatomik mod¨ullerin ¨ozellikleri c¸alıs¸ılmıs¸tır. Bir M mod¨ul¨u w-t¨umlenmis¸ mod¨uld¨ur ancak ve ancak M bol w-t¨umlenmis¸ mod¨uld¨ur. W -t¨umlenmis¸ mod¨ullerin sınıfı belli bir s¸artı sa˘glamak kos¸ulu ile genis¸letme altında kapalı olmasına ra˘gmen, w-koatomik mod¨ullerin sınıfı genis¸letme altında her zaman kapalıdır. Bir w-t¨umlenmis¸ mod¨ul¨un her alt mod¨ul¨un¨un w-t¨umlenmis¸ olmadı˘gını g¨osteren bir ¨ornek verebiliriz. Dedekind tamlık b¨olgesi ¨uzerinde her burulmalı mod¨ul (t¨umden) w-t¨umlenmis¸ mod¨uld¨ur. E˘ger R yarılocal bir halka ise o zaman her R-mod¨ul (t¨umden) w-t¨umlenmis¸tir. Bir sol V -halkası ¨uzerinde w-t¨umlenmis¸ mod¨uller ile t¨umden w-t¨umlenmis¸ mod¨uller c¸akıs¸ıktır. E˘ger M indirgenmis¸ ve

w-t¨umlenmis¸ bir mod¨ul ise M w-koatomiktir.

Anahtar s¨ozc ¨ukler: Yarıbasit, radikal, socle, w-t¨umlenmis¸, t¨umden w-t¨umlenmis¸,

w-koatomik.

11 Page

Ph.D. THESIS EXAMINATION RESULT FORM ... ii

ACKNOWLEDGMENTS ... iii

ABSTRACT ... iv

¨ OZ ... v

CHAPTER ONE - PRELIMINARIES ... 1

1.1 Small and Large Modules ... 2

1.2 Simple and Semisimple Modules ... 3

1.3 Radical and Socle of Module... 4

1.4 Semilocal Modules ... 5

1.5 Torsion Modules ... 6

1.6 Injective Modules ... 7

1.7 Supplemented Modules ... 7

1.8 Totally Supplemented Modules ...10

1.9 Coatomic Modules...12

CHAPTER TWO - W-SUPPLEMENTED MODULES ...14

2.1 W-Supplemented Modules...14

2.2 W-Supplemented Modules over Commutative Rings...22

2.3 Totally W-Supplemented Modules ...25

CHAPTER THREE - W-COATOMIC MODULES ...31

3.1 W-Coatomic Modules ...31

CHAPTER FOUR - CONCLUSION ...40

REFERENCES ...41

NOTATION ...44 INDEX ...45

PRELIMINARIES

Since Kasch and Mares have transferred the notions of perfect and semiperfect rings to modules, the notion of supplemented module used extensively by many authors. A module M is said to be supplemented, if for every submodule A of M, there exists a submodule B of M such that M = A + B and A∩ B ≪ B. A module M is said to be amply supplemented, if whenever M = A + B then B contains a supplement of A and a module M is weakly supplemented, if there is a submodule B of M such that

M = A + B and A∩ B ≪ M. In early years, supplemented modules and two other

generalizations amply supplemented and weakly supplemented modules are studied by Helmut Z¨oschinger and he posed their whole structure over discrete valuation rings. After Z¨oschinger, some variations of supplemented modules are studied. P.F. Smith and D. Keskin studied on ⊕-supplemented modules and also studied by R. Tribak and A. Idelhadj. While cofinitely supplemented modules are studied by R. Alizade, P.F. Smith and G. Bilhan also cofinitely weak supplemented modules studied by R. Alizade and E. B¨uy¨ukas¸ık, totally and totally cofinitely supplemented modules by P.F. Smith and G. Bilhan. In recent years, rad-supplemented modules are studied by W. Yongduo and D. Nanging and cofinitely rad-supplemented modules by E. B¨uy¨ukas¸ık and C. Lomp.

It is well-known that a supplement submodule in supplemented module need not be a direct summand. But we can state that for a semisimple submodule U of a module M, supplement submodule of U is a direct summand in M. If every semisimple submodule of a module M has a direct summand supplement, we say that M is

w-supplemented module. In this chapter we introduce some basic terminology for

modules and fundamental properties which they will used in the following sections. Throughout this thesis all rings are associative with identity and all modules are unitary

left modules.

In second chapter, we study w-supplemented and totally w-supplemented modules. We show by an example that a submodule of a w-supplemented module need not be

w-supplemented. Under a certain condition we prove that the class of w-supplemented

modules is closed under extension by short exact sequence. We say that a module

M is w-supplemented if and only if M is amply w-supplemented. We determine the

relationships between w-supplemented and totally w-supplemented modules over a left

V -ring.

Finally in third chapter, we deal with w-coatomic modules. A module M is

w-coatomic in terms of a w-coatomic submodule L and w-coatomic factor module M

L. It is proved that if a module M is reduced and w-supplemented module, then M is w-coatomic. Over a discrete valuation ring, every submodule of Rad(M) is w-coatomic

if and only if every submodule U of M is w-coatomic.

1.1 Small and Large Modules

Definition 1.1.1. Let R be a ring and M be an R-module. A submodule K of M is called superfluous or small in M, written K≪ M, if, for every submodule L of M, the equality K + L = M implies L = M. For an R-module N, an epimorphism f : M→ N is said to be small cover if Ker f ≪ M.

Definition 1.1.2. A non-zero submodule K of a module M is essential (or large) in M, abbreviated KE M, in case for every submodule L of M, K ∩ L = 0 implies L = 0. Lemma 1.1.3. (by Wisbauer (1991,§19.3 & §19.6)) Let K, L, N and M be R-modules.

(i) If f : M→ N and g : N → L are two epimorphisms, then g◦ f is small if and only if f and g are small.

(ii) If K≤ L ≤ M, then L ≪ M if and only if K ≪ M and L K ≪

M K.

(iii) If K1, ..., Knare small submodules of M, then K1+ ... + Knis also small in M.

(iv) For K ≪ M and f : M → N we get f (K) ≪ N. The converse is true if, f is a small epimorphism.

(v) If K≤ L ≤ M and L is a direct summand in M, then K ≪ M if and only if K ≪ L. (vi) If K≪ M, then M is finitely generated if and only if M

K is finitely generated.

1.2 Simple and Semisimple Modules

Definition 1.2.1. A module M is called simple in case M̸= 0 and it has no nontrivial submodule. A module M is said to be semisimple if it is a direct sum of simple submodules.

Theorem 1.2.2. (by Anderson & Fuller (1992, Theorem9.6)) For a left R-module M,

the following statements are equivalent:

(i) M is semisimple;

(ii) M is generated by simple modules;

(iii) M is the sum of some set of simple submodules; (iv) M is the sum of its simple submodules;

(v) Every submodule of M is a direct summand; (vi) Every short exact sequence

0→ K → M → N → 0

Corollary 1.2.3. (by Kasch (1982, Corollary8.1.5))

(i) Every submodule of a semisimple module is semisimple.

(ii) Every epimorphic image of a semisimple module is semisimple. (iii) Every sum of semisimple modules is semisimple.

1.3 Radical and Socle of Module

Definition 1.3.1. For an R-module M and N we have

Rad(M) =∩{K ≤ M | K is maximal in M}

=∑{L ≤ M | L is small in M}

Lemma 1.3.2. (by Wisbauer (1991, §21.6) & Kasch (1982, 9.1.5)) For an R-module

M and N, we have the following properties:

(i) For a morphism f : M→ N we have • f (RadM) ⊂ RadN.

• Rad( M

RadM) = 0.

• f (RadM) = Rad( f (M)) if Ker f ⊂ RadM.

(ii) For an epimorphism φ : M → N where Kerφ ≪ M, we have

φ(Rad(M)) = Rad(N) and Rad(M) = φ−1_(Rad(N))

(iii) If every submodule of M is contained in a maximal submodule, then RadM≪ M. (iv) M is finitely generated if and only if RadM≪ M and M

(v) If M =⊕_∧M_λ, then RadM =⊕_∧RadM_λand M RadM ∼= ⊕ ∧_RadMMλ λ forλ ∈ ∧. (vi) If M

RadM is semisimple and RadM≪ M, then every proper submodule of M is contained in a maximal submodule.

(vii) For a submodule K of M, in general, RadK̸= K ∩ RadM. (viii) If C is a submodule of M, then Rad(C) is submodule of Rad(M).

Definition 1.3.3. Let M be an R-module. The socle of M, denoted by Soc(M) or SocM, is the sum of all simple submodules of M, equivalently, intersection of all essential submodules of M. Note that M is semisimple if and only if Soc(M) = M.

Lemma 1.3.4. (by Wisbauer (1991, §21.2) & Kasch (1982, 9.1.5)) Let M and N be

R-modules. Then we have the following properties:

(i) For any morphism f : M→ N, f (SocM) ⊂ SocN.

(ii) For a monomorphism φ : M → N where Imφ E N, φ(Soc(M)) = Soc(N) and Soc(M) =φ−1(Soc(N)).

(iii) For any submodule K of M, SocK = K∩ SocM.

(iv) If C is a submodule of M, then Soc(C) is submodule of Soc(M). (v) Soc(⊕_∧M_λ) =⊕_∧Soc(M_λ) forλ ∈ ∧.

(vi) Soc(Soc(M)) = Soc(M).

1.4 Semilocal Modules

Definition 1.4.1. A module M is called semilocal if M

Rad(M) is semisimple. A ring R

is said to be semilocal provided R

Rad(R) is a left (or right) semisimple ring. A ring R

Proposition 1.4.2. (by Lomp (1999, Proposition2.1)) For a proper submodule N of a

module M, the followings are equivalent:

(i) M

N is semisimple;

(ii) For every submodule L of M, there exists a submodule K of M such that L + K = M and L∩ K ≤ N;

(iii) There exists a decomposition M = M1⊕ M2such that M1is semisimple, NE M2

and M2

N is semisimple.

Proposition 1.4.3. (by Clark et al. (2006, §17.3)) The class of semilocal modules is

closed under factor modules, direct sums and small covers.

1.5 Torsion Modules

Definition 1.5.1. A submodule T (M) ={m ∈ M : rm = 0 for some 0 ̸= r ∈ R} of a module M is called torsion submodule of M. If T (M) = M, then M is said to be a

torsion and if T (M) = 0, then M is called a torsion-free module. Let P be a prime ideal

of R. The submodule{m ∈ M : Pnm = 0 for some n≥ 1} is said to be P-primary part

of M and denoted by TP(M).

Theorem 1.5.2. (by Cohn (2002, Proposition10.6.9)) Any torsion module M over a

Dedekind domain is a direct sum of its primary parts, in a unique way:

M =⊕

P∈Ω

TP(M)

whereΩ is the set of all maximal ideals of a ring R.

module over a Dedekind domain. Then

M = T (M)⊕ P

where P is a torsion-free submodule of M and T (M) is the torsion submodule of M.

1.6 Injective Modules

Definition 1.6.1. A module E is injective for every monomorphism f : A→ B and homomorphism g : A→ E, there exist a homomorphism h : B → E such that h ◦ f = g. Lemma 1.6.2. (by Sharpe & Vamos (1972, Theorem 2.15)) For an R-module E, the

following statements are equivalent: a) E is injective.

b) E is a direct summand of every extension of itself.

1.7 Supplemented Modules

Definition 1.7.1. Let U be a submodule of a module M. A submodule V of M is called a supplement of U in M, if V is a minimal element in the set of submodules L of M with U + L = M. Equivalently, V is a supplement of U if and only if U + V = M and

U∩V ≪ V.

Lemma 1.7.2. (by Z¨oschinger (1974a, Lemma1.2)) Let M be an R-module and U,V

be submodules of M. Let V be a supplement of U in M. Then

(i) X  V ⇒V X ̸≪

M X

(iii) RadV = V∩ RadM (iv) Rad(M

U) =

RadM +U U

(v) RadM = (V + RadM)∩ (U + RadM) = (V ∩ RadM) + (U ∩ RadM)

Proposition 1.7.3. (by Wisbauer (1991, §41.1)) Let U,V be submodules of an

R-module M and let V be a supplement of U in M. Then:

(i) If M is finitely generated, then V is also finitely generated.

(ii) If U is a maximal submodule of M, then V is cylic, and U∩V = Rad(V) is a (the unique) maximal submodule of V .

(iii) If Rad(M)≪ M, then U is contained in a maximal submodule of M. (iv) For L≤ U, V + L L is a supplement of U L in M L.

(v) If Rad(M) ≪ M or Rad(M) ≤ U, and if p : M → M

Rad(M) is the canonical projection, then M

Rad(M) = p(U )⊕ p(V).

Definition 1.7.4. An R-module M is called supplemented module, if every submodule of M has a supplement in M.

Lemma 1.7.5. (by Wisbauer (1991,§41.2)) Let M be an R-module. Then we have the

following statements:

(i) Let M1,U be submodules of M with M1 supplemented module. If there is a

supplement for M1+U in M, then U also has a supplement in M.

(ii) If M = M1+ M2 where M1 and M2 are supplemented modules, then M is also

supplemented module.

(iii) If M is supplemented module, then:

a) Every finitely M-generated module is supplemented, b) M

Theorem 1.7.6. (by Z¨oschinger (1974a, Theorem 2.4)) Let R be a discrete valuation

ring. A module M is supplemented if and only if M = M1⊕ M2⊕ M3⊕ M4 where

M1∼= Rn1, M2∼= Kn2, M3∼= (K/R)n3 and pn4M4= 0 for some integer ni≥ 0 and prime

p.

Theorem 1.7.7. (by Z¨oschinger (1974a, Folgerung p.51)) For a discrete valuation

ring, the following are equivalent:

(i) Every submodule of a supplemented module is supplemented,

(ii) Every extension of a supplemented module by a supplemented module is itself supplemented,

(iii) If every submodule of the radical of a module M has a supplement, then M is supplemented,

(iv) Every torsion free reduced module with finite rank is free.

Theorem 1.7.8. (by Z¨oschinger (1974a, Theorem3.1)) Let R be a non-local Dedekind

domain. An R-module M is supplemented if and only if it is torsion and every primary part is supplemented.

Definition 1.7.9. A submodule N of a module M is called cofinite if M

N is finitely

generated. Let us call a module M cofinitely supplemented, if every cofinite submodule of M has a supplement in M. Obviously, finitely generated cofinitely supplemented module is supplemented.

Definition 1.7.10. We say a submodule N of a module M has ample supplement in M, if for every submodule L of M with M = N + L , there is a supplement L′ of N with

L′≤ L. A module M is called amply supplemented module, if every proper submodule

of M has an ample supplement in M.

Lemma 1.7.11. (by Wisbauer (1992, §41.7)) Let M be an amply supplemented

(i) Every supplement of a submodule of M is an amply supplemented module. (ii) Direct summands and factor modules of M are amply supplemented.

Proposition 1.7.12. (by Clark et al. (2006, §20.24)) The following properties are

equivalent for a module M:

(i) M is amply supplemented module;

(ii) Every submodule U of M is of the form U = X + Y , with X a supplement and Y ≪ M;

(iii) For every submodule U of M, there is a supplement submodule X of U such that X is cosmall in M.

If M is finitely generated, then (i)− (iii) are also equivalent to: (iv) Every maximal submodule has ample supplements in M.

1.8 Totally Supplemented Modules

Definition 1.8.1. A module M is called totally (cofinitely) supplemented, if every submodule of M is (cofinitely) supplemented.

Theorem 1.8.2. (by Bilhan (2007, Theorem 1)) The following statements are

equivalent:

(i) R is a semiperfect ring;

(ii) Every left (right) R-module is cofinitely supplemented; (iii) Every left (right) R-module is amply cofinitely supplemented;

Theorem 1.8.3. (by Bilhan (2007, Theorem 16)) Let R be a commutative noetherian

ring and M be a coatomic R-module. The following statements are equivalent:

(i) M is cofinitely supplemented; (ii) M is amply cofinitely supplemented; (iii) M is totally cofinitely supplemented.

Corollary 1.8.4. (by Bilhan (2007, Corollary18)) If R is a commutative domain, then

every totally cofinitely supplemented R-module is torsion.

Lemma 1.8.5. (by Smith (2000, Corollary1.2)) Every totally supplemented module is

amply supplemented and every amply supplemented module is supplemented.

Theorem 1.8.6. (by Smith (2000, Theorem 1.3)) Let R be a non-local Dedekind

domain. Then the following statements are equivalent for an R-module M.

(i) M is supplemented; (ii) M is amply supplemented; (iii) M is totally supplemented;

(iv) M is a torsion module such that Tp(M) is a direct sum of an Artinian submodule

and a bounded submodule, for each maximal ideal p of R.

Theorem 1.8.7. (by Smith (2000, Theorem 2.8)) Let K be a linearly compact

submodule of a module M. Then M is totally supplemented if and only if M

K is totally supplemented.

Theorem 1.8.8. (by Smith (2000, Theorem2.9)) Let a module M = M1⊕M2be a direct

sum of submodules M1, M2such that M2is semisimple. Then M is totally supplemented

Lemma 1.8.9. (by Smith (2000, Lemma 4.2)) Let R be a commutative ring and a

module M = M1⊕ ... ⊕ Mn be a finite direct sum of totally supplemented submodules

Miwhere 1≤ i ≤ n, for some positive integer n ≥ 2, such that R = ann(Mi) + ann(Mj)

for all 1≤ i < j ≤ n. Then M is totally supplemented.

Theorem 1.8.10. (by Smith (2000, Theorem4.7)) Let R be a commutative ring. Then

every Noetherian supplemented R-module is totally supplemented.

1.9 Coatomic Modules

Definition 1.9.1. Let M be an R-module. M is called coatomic module if Rad(M

U) = M U

implies M = U for every submodule U of M, or equivalently, every proper submodule of M is contained in a maximal submodule of M.

Definition 1.9.2. A submodule N of a module M is said to be radical, if Rad(N) = N. Let P(M) =∑{N ≤ M : Rad(N) = N}. The module M is called reduced, if P(M) = 0. Lemma 1.9.3. (by Z¨oschinger (1974a, Lemma1.3))

(i) Let A, B,C be R-modules and

0→ A → B → C ∼= B/A→ 0

be exact sequence. If A and C are reduced (coatomic) module, then B is also reduced (coatomic) module.

(ii) Let U,V be submodules of a module M and let V be a supplement of U in M. Then M

U is radical (coatomic) module if and only if V is radical (coatomic) module.

(i) Every factor module of a coatomic module M is coatomic.

(ii) Every extension of a coatomic module by a coatomic module is coatomic.

Lemma 1.9.5. (by G¨ung¨oro˘glu (1998, Lemma10)) Let M be a supplemented module.

If M is reduced or Rad(M) is small in M, then M is coatomic.

Theorem 1.9.6. (by G¨ung¨oro˘glu (1998, Theorem 12)) Let M be a supplemented

module such that Rad(M) is coatomic module. Then M is coatomic module.

Lemma 1.9.7. (by G¨ung¨oro˘glu (1998, Lemma 11)) Let M be a module and U be a

submodule of M such that U ⊆ Rad(M). If U is coatomic, then U is small in M.

Lemma 1.9.8. (by Z¨oschinger (1974a, Lemma 2.1)) Over a discrete valuation ring,

for an R-module M, the following are equivalent:

(i) M has a small radical; (ii) M is coatomic;

(iii) M is a direct sum of a finitely generated and a bounded submodule; (iv) M is reduced and supplemented.

Lemma 1.9.9. (by G¨ung¨oro˘glu (1998, Corollary 5)) Let M be an R-module and let M =⊕n_i=1Mibe a finite direct sum of submodules Mi(i = 1, ..., n). Then M is coatomic

if and only if each Miis coatomic.

Lemma 1.9.10. (by Z¨oschinger (1980, Lemma 1.1)) Let M be an R-module and let M

be a coatomic module over commutative Noetherian ring. Then every submodule of M is coatomic.

W-SUPPLEMENTED MODULES

2.1 W-Supplemented Modules

In this chapter, we deal with w-supplemented modules and totally w-supplemented modules. For more details about supplemented modules see the book Wisbauer (1991). Definition 2.1.1. (Keskin & Orhan, 2006) A module M is said to be a weak

⊕-supplemented module, if for each semisimple submodule N of M, there exists a

direct summand K of M such that M = N + K and N∩ K ≪ K.

Definition 2.1.2. (Keskin & Orhan, 2006) A module M is weak lifting, if for each semisimple submodule N of M, there exists a decomposition M = M1⊕ M2 such that

M1≤ N and M2∩ N ≪ M2.

Proposition 2.1.3. (by Keskin & Orhan (2006, Proposition2.3)) A weak lifting module

is weak-⊕-supplemented.

By inspiring from these definitions, the following definition can be given.

Definition 2.1.4. A module M is w-supplemented, if every semisimple submodule of

M has a supplement in M.

Lemma 2.1.5. If M is w-supplemented module, then M = N⊕ S for some semisimple

submodule N and a submodule S of M.

Proof. Let K be a semisimple submodule of M. If there is no non-zero K, then M = 0⊕

M and result follows. Otherwise, by assumption, K has a supplement S, i.e., M = K + S

and K∩ S ≪ S. Obviously, K ∩ S is a direct summand of K, because K is semisimple. So, K = (K∩S)⊕X for some submodule X of K. Then M = [(K ∩S)⊕X]+S = X +S but X∩ S = (K ∩ S) ∩ X = 0. So, M = X ⊕ S.

The following is an immediate consequence of the above result.

Corollary 2.1.6. Let M = N + L where L is a submodule of M and N is a semisimple

submodule of M. Then M = N′⊕ L for some submodule N′of N.

Proof. Let N be a semisimple submodule of M. Then N∩ L is direct summand in N.

That is, N = (N∩L)⊕N′for some submodule N′of N. Since M = N + L, then we have

M = ((N∩L)⊕N′)+L = N′+L. So M = N′⊕L because (N ∩L)∩N′= N′∩L = 0.

We proved that, there is no need a discretion of all these above three definitions. Theorem 2.1.7. For an R-module M, the following statements are equivalent: (1) M is w-supplemented;

(2) M is weak⊕-supplemented; (3) M is weak lifting.

Proof. (1)⇒ (2) Let N be a semisimple submodule of M. Since M is w-supplemented, N has a supplement L in M such that M = N + L and N∩L ≪ L. From Corollary 2.1.6, M = N′⊕ L for some submodule N′ of N. Therefore N is the desired semisimple submodule mentioned in the definition of weak-⊕-supplemented module.

(2)⇒ (3) Let M be weak-⊕-supplemented module and N be semisimple submodule of M. Then N has a direct summand supplement K in M, i.e., M = N + K = K⊕L′and

N∩ K ≪ K for some submodule L′of M. Since N is semisimple, by Corollary 2.1.6, then M = N′⊕ K for some submodule N′of N. Hence M is weak lifting, because K is

M2and N′is M1of definition 2.1.2.

(3)⇒ (1) Let N be a semisimple submodule of M. Since M is weak lifting, there exists a decomposition of M such that M = K⊕ K′, K ≤ N and N ∩ K′≪ K′for some submodule K′ of M. Then M = N + K′. This means that N has a supplement K′in M.

Thus M is w-supplemented.

After this theorem, we shall drop the other names and call all these modules by

w-supplemented modules.

We could generalize the definition of w-supplemented module as follows:

Proposition 2.1.8. A module M is w-supplemented if and only if for every semisimple

submodule N of M, M = N + L and N∩ L ≪ M for some submodule L of M.

Proof. (⇒) Let N be a semisimple submodule of M. Since M is w-supplemented, there

exists a supplement L in M such that M = N + L and N∩ L ≪ L. By Lemma 1.1.3,

N∩ L ≪ M.

(⇐) Let K be a semisimple submodule of M. By assumption, there exists a submodule

L of M such that M = K + L and K∩L ≪ M. Then M = K′⊕L for some submodule K′

of K by Corollary 2.1.6. By Lemma 1.1.3, K∩ L ≪ L. Hence M is w-supplemented.

We should point out that, analog of the above proposition for supplemented modules is not true. It is the concept of weakly supplemented modules (see [6]).

Example 2.1.9. The Z-module Z of integers is w-supplemented, because the only semisimple submodule ofZ is the zero submodule but obviously it is not supplemented. Lemma 2.1.10. Let M be an R-module. If Soc(M)≪ M, then M is w-supplemented.

Proof. Let N be semisimple submodule of M. Since Soc(M) is the largest semisimple

submodule of M, then N is submodule of Soc(M). Since Soc(M)≪ M, by Lemma 1.1.3 N ≪ M. Then M = M + N and M ∩ N = N ≤ Soc(M) ≪ M. Hence N has a supplement in M.

Lemma 2.1.11. (by Harmanci & Keskin & Smith (1999, Lemma1.3)) Let N and L be

submodules of a module M such that N + L has a supplement H in M and N∩ (H + L) has a supplement G in N. Then H + G is a supplement of L in M.

Proposition 2.1.12. Finite direct sum of w-supplemented modules is w-supplemented.

Proof. Let n be any positive integer and let Mi be w-supplemented module for each

1≤ i ≤ n. Let M = M1⊕ M2⊕ ... ⊕ Mn. It is sufficient to prove by induction on n.

Suppose n = 2. Then M = M1⊕M2. For i = 1, 2, let pi: M→ Mibe the projection map.

Let L be a semisimple submodule of M. Then M = M1+ M2+ L so that M1+ M2+ L

has a supplement 0 in M. By Corollary 1.2.3, p2(L) = M2∩ (M1+ L) is semisimple.

Then M2∩ (M1+ L) has a supplement H in M2 such that H is a direct summand of

M2. By Lemma 2.1.11, H is a supplement of M1+ L in M. Since H⊆ M2, it implies

that (L + H)∩ M1⊆ (L + M2)∩ M1= p1(L). So again by Corollary 1.2.3, p1(L) is

semisimple. Thus (L + H)∩ M1has a direct summand supplement K in M1. Again by

Lemma 2.1.11, H + K is a supplement of L in M. Hence M is w-supplemented. Theorem 2.1.13. For an R-module M, the following statements are equivalent: (1) M is w-supplemented module;

(2) Soc(M) has a supplement in M.

Proof. (1)⇒ (2) Let M be w-supplemented. Then every semisimple submodule of M

has a supplement in M. Since Soc(M) is the largest semisimple submodule, Soc(M) has a supplement in M.

V∩ Soc(M) ≪ V. By Lemma 2.1.10, V is w-supplemented. Since every finite direct

sum of w-supplemented modules is w-supplemented, M is w-supplemented.

The following result is proved in (Keskin & Orhan (2006), Lemma 2.4) by adding an extra property, namely (D3) property, however (D3) property is unnecassary. Lemma 2.1.14. (by Keskin & Orhan (2006, Lemma2.4)) If M is a w-⊕-supplemented

module satisfying (D3), then any direct summand of M is w-⊕-supplemented.

Proposition 2.1.15. Any direct summand of a w-supplemented module is

w-supplemented.

Proof. Let M be w-supplemented module and N be a direct summand of M so that M = N⊕ N′ for some submodule N′ of M. Let K be a semisimple submodule of

N. Since K is also a semisimple submodule of M, then we have M = K + K′ and

K∩ K′≪ K′ for some submodule K′ of M. By Lemma 2.1.5, we get M = L⊕ K′ for some submodule L of K. By the modular law, N = K + (N∩ K′) = L⊕ (N ∩ K′). We write N∩ (K ∩ K′)⊆ K ∩ K′≪ K′, so by Lemma 1.1.3, N∩ (K ∩ K′)≪ K′ and also

N∩ (K ∩ K′)≪ M. Since N is direct summand in M, N ∩ (K ∩ K′)≪ N by Lemma 1.1.3. It follows from Lemma 1.1.3 that N∩ (K ∩ K′) = K∩ K′≪ N ∩ K′. It means that K has a supplement N∩ K′in N. Therefore N is w-supplemented.

Proposition 2.1.16. Any small cover of a w-supplemented module M is w-supplemented.

Proof. Let M be w-supplemented and N be a small cover of M, then there exists an

epimorphism f : N→ M with Ker f ≪ N. Let L be a semisimple submodule of N. By Corollary 1.2.3, f (L) is semisimple in M, then by assumption, f (L) has a supplement

X in M for some submodule X in M. That is, M = f (L) + X , X∩ f (L) ≪ X and from

Corollary 2.1.6, for some submodule Y of f (L), M = X⊕Y. M = X + f (L) implies that N = L + f−1(X ) and since X∩ f (L) ≪ X, it follows f−1(X )∩ L ≪ f−1(X ).

Lemma 2.1.17. Let

0→ L → M → N → 0

be a short exact sequence for R-modules. If L and N are w-supplemented and L≪ M, then M is w-supplemented.

Proof. Let us consider N as M

L. Let U be a semisimple submodule of M. Then U + L L is semisimple submodule in M L. If M L = U + L L , then M = U + L. Since U

is semisimple and L is w-supplemented, then M is w-supplemented as a finite sum of

w-supplemented modules. Let U + L

L be a proper submodule of M L. By assumption, U + L L has a supplement V L in M L. That is, M L = (U + L) L + V L and (U + L) L ∩ V L ≪ V L. Therefore M = U +V and(U∩V) + L L ≪ V L. By Corollary 2.1.6, M = U ′_{⊕V for some}

submodule U′of U . Let us show U∩V ≪ V. Let V = (U ∩V)+X for some submodule

X of V . ThenV L = (U∩V) + L L + X + L L . Since (U∩V) + L L ≪ V L, so V L = X + L L . It

follows that V = X + L. By Lemma 1.1.3, if L≪ M, then L ≪ V. Hence V = X. Lemma 2.1.18. (by Lomp & Wisbauer & Clark & Vanaja (2006, §2.8.(9))) Let M

be an R-module and let Rad(M) be radical of M and Soc(M) be socle of M. Then Soc(Rad(M))≪ M.

Lemma 2.1.19. Let M be a module and U be a semisimple submodule of M contained

in Rad(M). Then U is small in M.

Proof. Let U⊆ Rad(M) where U is semisimple in M. By Lemma 1.3.4, it follows that Soc(U )⊆ Soc(Rad(M)). By Definition 1.3.3, Soc(U) = U. Then U ⊆ Soc(Rad(M)).

By Lemma 2.1.18 and Lemma 1.1.3, U ≪ M.

Corollary 2.1.20. Every radical module is w-supplemented.

2.1.18, Soc(M) = Soc(Rad(M))≪ M. Thus by Lemma 2.1.10, M is w-supplemented.

Clearly, every semisimple module is w-supplemented. As we will see the following example, the converse is not true in general.

Example 2.1.21. Let M =Z_(p∞₎ whereZ_(p∞₎ is the Pr¨ufer p-group for any prime p. Then Soc(M)̸= 0 because Soc(Z_(p∞₎) = Z

pZ. Since Rad(Z(p∞)) =Z(p∞), Z(p∞) is a

radical module. By Corollary 2.1.20, M is a w-supplemented module. But M is not semisimple.

Proposition 2.1.22. Let M

Rad(M) be semisimple, then M is w-supplemented module.

Proof. Let N be a semisimple submodule of M. Then N + Rad(M)

Rad(M) is a semisimple submodule of M Rad(M). If N + Rad(M) Rad(M) = M

Rad(M), then M = N + Rad(M). Thus M = N′⊕ Rad(M) for some submodule N′ of N by Corollary 2.1.6. Since N∩ Rad(M) is semisimple, then Soc(N∩Rad(M)) = N ∩Rad(M) and so N ∩Rad(M) ⊆ Rad(M). By Lemma 2.1.19, N∩ Rad(M) ≪ M and also by Lemma 1.1.3, N ∩ Rad(M) ≪ Rad(M) because Rad(M) is direct summand in M. Let N + Rad(M)

Rad(M) be a proper semisimple

submodule of M

Rad(M). By assumption,

N + Rad(M)

Rad(M) is direct summand in M Rad(M), that is, M Rad(M) = N + Rad(M) Rad(M) ⊕ N′

Rad(M) for some submodule N

′ _{of M. Therefore}

M = N + N′such that N∩ N′⊆ Rad(M) and by Corollary 2.1.6, M = K ⊕ N′for some submodule K of N. Since N∩ N′ is semisimple, by Lemma 2.1.19, N∩ N′≪ M. If

N∩N′≪ M, by Lemma 1.1.3, N ∩N′≪ N′. That is, N has supplement N′in M. Hence

M is w-supplemented.

Proposition 2.1.23. A module M is w-supplemented if and only if M is amply

Proof. (⇐) Clear.

(⇒) Let M = A + B where A is semisimple. Since M is w-supplemented, by Lemma 2.1.5, M = A′⊕C where A′⊆ A and A ∩C ≪ C. By the modularity, A = A′⊕ (A ∩C), then M = A + B = A′⊕ (A ∩C) + B and then by Lemma 1.1.3, A ∩C ≪ M, therefore

M = A′+ B, because of the fact that M = A′⊕ C, B can be considered as a direct summand of M, so M = A′⊕ B. Since A is semisimple, then so is A ∩ B. If A ∩ B = 0,

M is immediately amply w-supplemented. Let A∩B ̸= 0. By Lemma 2.1.5, M =Y1⊕T

for some submodule Y1 of A∩ B and T is supplement of A ∩ B in M. By modular law,

A∩ B = Y1⊕ (A ∩ B ∩ T). Let us call A ∩ B ∩ T = S, by applying the modular law,

again to M = Y1⊕ T, we get B = Y1⊕ (B ∩ T). Let us call B ∩ T = Y2. We consider the

projection mappingπ2: Y1⊕Y2→ Y2, then

A∩ B = Y1⊕ S = Y1⊕ (T ∩ (A ∩ B)) = Y1⊕ (A ∩Y2) = Y1⊕ (A ∩Y2∩ B)

since Y2⊆ B. Then A ∩Y2= A∩ B ∩Y2=π2(A∩ B) = π2(Y1+ S) =π2(S)≪ Y2, since

S≪ T. Also M = A + B = A + Y1+ Y2= A + Y2. Therefore A has supplement Y2

contained in Y .

The following Corollary is consequence of Proposition 2.1.22 and Proposition 2.1.23.

Corollary 2.1.24. Over a semilocal ring, all R-modules are (amply) w-supplemented.

Proof. Let R be a semilocal ring and M be an R-module. Then M

Rad(M) is semisimple.

By Proposition 2.1.22 and Proposition 2.1.23, M is (amply) w-supplemented.

Definition 2.1.25. (by Turkmen & Pancar, (2009) or by Yongduo & Nanging, (2006)) Let M be an R-module and let A be a submodule of M. A submodule B of M is called a radical supplement (or briefly Rad-supplement) of A in M (according to Yongduo & Nanging (2006), generalized supplement), if A + B = M and A∩ B ⊆ Rad(B). An

R-module M is said to be radical supplemented (or briefly Rad-supplemented) if every

submodule of M has a Rad-supplement in M (according to Yongduo & Nanging (2006), generalized supplemented module).

Proposition 2.1.26. A module M is w-supplemented if and only if every semisimple

submodule of M has a Rad-supplement in M.

Proof. (⇒) Let M be w-supplemented and N be a semisimple submodule of M. By

assumption, N has a supplement K in M. That is, M = N + K and N∩ K ≪ K. Then

N∩ K ⊆ Rad(K). Thus K is Rad supplement of N in M.

(⇐) Let N be a semisimple submodule of M. By assumption, N has Rad-supplement

K in M. Then M = N + K and N∩ K ⊆ Rad(K). By Corollary 2.1.6, then M = N′⊕ K for some submodule N′ of N. If N is semisimple, then so is N∩ K. Since

N∩ K ⊆ Rad(M), by Lemma 2.1.19, Soc(N ∩ K) = N ∩ K ⊆ Soc(Rad(M)) ≪ M, that

is, N∩ K ≪ M. By Lemma 1.1.3, N ∩ K ≪ K. Hence N has supplement K in M. Thus

M is w-supplemented.

2.2 W-Supplemented Modules over Commutative Rings

Lemma 2.2.1. Let R be a discrete valuation ring and let M be an R-module. If N is a

submodule of M such that N∩ Rad(M) = 0, then N is w-supplemented.

Proof. In a discrete valuation ring, M

Rad(M) is semisimple because unique maximal

submodule of M is Rad(M) in M. By the isomorphism theorem, N + Rad(M)

Rad(M) ∼= N

N∩ Rad(M) ∼= N is submodule of M

Rad(M), then N is semisimple. Therefore N is w-supplemented.

maximal ideal of R, then for every P-primary R-module M, M

Rad(M) is semisimple.

Proof. Since R is commutative, we have

RadM = ∩

Q∈Ω

QM.

First we will show that QM = M for every Q∈ Ω\{P}. Let x ∈ M, then Pnx = 0 for

some n∈ N. Since Pn+ Q = R, we have 1 = p + q for some p∈ Pnand q∈ Q. So we get x = xp + xq = xq∈ QM, hence M = QM. Therefore

RadM = ∩

Q∈Ω

QM = PM.

Then M/RadM = M/PM is a semisimple R/P-module since R/P is a field, and so it is semisimple as an R-module.

Corollary 2.2.3. (by B¨uy¨ukas¸ık (2005, Corollary4.1.2)) Let R be a Dedekind domain

and M be a torsion R-module, then M

Rad(M) is semisimple.

Proof. Since R is a Dedekind domain and M is a torsion R-module, we have

M =⊕ P∈Ω TP(M). Then M/RadM = [⊕ P∈Ω TP(M)]/[ ⊕ P∈Ω Rad(TP(M))] ∼= ⊕ P∈Ω [TP(M)/Rad(TP(M))] is semisimple by Lemma 2.2.2.

Theorem 2.2.4. Let R be a Dedekind domain and M be a torsion R-module. Then M

Proof. By Corollary 2.2.3, M

Rad(M) is semisimple, and by Proposition 2.1.22, M is w-supplemented.

Lemma 2.2.5. (by Anderson & Fuller (1992, Lemma 5.19)) A submodule K of a

module M is essential in M if and only if for each 0̸= x ∈ M there exists an r ∈ R such that 0̸= rx ∈ K.

Lemma 2.2.6. Let R be a Dedekind domain. If Rad(M) E M, then M is w-supplemented.

Proof. Let Rad(M)E M. To prove M

Rad(M) is torsion we will show that there exists

a non-zero element r of R such that r(m + Rad(M)) = 0, that is, rm∈ Rad(M). Then for m∈ M, m+Rad(M) ∈ M

Rad(M). Since Rad(M)EM, by Lemma 2.2.5, there exists

an element r of R such that 0̸= rm ∈ Rad(M). Thus M

Rad(M) is torsion. By Corollary

2.2.3, M

Rad(M) is semisimple. By Proposition 2.1.22, M is w-supplemented.

Lemma 2.2.7. (by Clark et al. (2006, §2.9)) Let M be an R-module such that Rad(M)E M. Let K ≤ L ≤ M be submodules of M and assume K to be a direct summand of M. Then Rad(K) = Rad(L) if and only if K = L.

Lemma 2.2.8. Let M be an R-module such that Rad(M)E M. Then the following

statements are equivalent:

(1) M is w-supplemented;

(2) Every semisimple submodule of M is a direct summand; (3) Soc(M) is a direct summand of M.

Proof. (1)⇒ (2) Let N be a semisimple submodule of M. By (1), N has a supplement K in M. Then M = N + K and N∩ K ≪ K. By Corollary 2.1.6, M = N′⊕ K for some

submodule N′ of N. Since Rad(N) = Rad(N′) = 0, by Lemma 2.2.7, N = N′. Thus

(2)⇒ (3) Since Soc(M) is semisimple submodule of M, Soc(M) is direct summand of

M by (2).

(3)⇒ (1) Let N be a semisimple submodule of M. Then N is a submodule and a direct summand of Soc(M). Since Soc(M) is a direct summand of M, N is direct summand of M. Then M is w-supplemented.

2.3 Totally W-Supplemented Modules

Definition 2.3.1. We say that a module M totally w-supplemented, if every submodule of M is w-supplemented.

Probably, it is expected that w-supplemented modules are also totally

w-supplemented. But unfortunately, it is not the case that converse is not true in

general.

In here, I would like to thank Christian LOMP for the following example.

Let R be a commutative ring and M be an R-module. Then it is not difficult to check that S = {  a m 0 a    : a ∈ R,m ∈ M }

is a ring with ordinary addition and multiplication.

Lemma 2.3.2. Let S be the ring giving above and R be a commutative ring. If M is a

faithful right R-module, then

Soc(SS) =   0 Soc(RM) 0 0    = {  0 m 0 0    ∈ S : m ∈ Soc(RM) }

and   0 M 0 0    ≪SS.

Proof. LetSI be a (non-zero) simple left ideal ofSS. Then I = S

  a m 0 a    for a ∈ R, m∈ M. Then _  0 M 0 0      a m 0 a    =   0 Ma 0 0    ≼ I

because I is an ideal, but since I is simple,   0 Ma 0 0    =   0 0 0 0   , then Ma = 0 ⇒ a = 0 or otherwise    0 Ma 0 0    = I = S    a m 0 a    and so,    a m 0 a    ∈ I which is

impossible. Hence I becomes S    0 m 0 0   , i.e., I = {   a m 0 a       0 m 0 0    : a ∈ R,m ∈ M } = {   0 am 0 0    : a ∈ R } =    0 Rm 0 0   

Thus Rm is a simple R-submodule of M, that is, I and consequently Soc(SS) is a

submodule of    0 Soc(M) 0 0   . Obviously    0 Soc(M) 0 0    ⊆ Soc(SS). Thus Soc(SS) =    0 Soc(M) 0 0   

For the other part: SinceSS is 2-generated, then Jac(S)≪SS and since

   0 M 0 0    2 =

   0 0 0 0   , then    0 M 0 0  

 ⊆ Jac(S) and so,    0 M 0 0    ≪SS.

Example 2.3.3. Let R be a commutative ring and M be a faithful right R-module with the properly that Soc(RM) has no supplement in M. For example, M =

∏

p−primeZp

is faithful but Soc(M) = ⊕_p−primeZp has no supplement in M where Zp’s are

Pr¨ufer Z-modules for various primes. Then SS =

{   a m 0 a    : a ∈ R,m ∈ M } is

w-supplemented but the submoduleSN =

   0 M 0 0  

 is not w-supplemented. Because,

by above lemma Soc(SS) =

   0 Soc(RM) 0 0  

 and Soc(SS) ≪S S. So, Soc(SS) has

supplement SS in SS.    0 Soc(RM) 0 0    is a semisimple submodule of    0 M 0 0   

but it has no supplement in SN, because if

   0 L 0 0  

 was a supplement of Soc(SS),

then obviously we will have Soc(RM) + L = M and Soc(RM)∩ L ≪ L, i.e., L is a

supplement of Soc(RM) in M, a contradiction.

Lemma 2.3.4. Let M be an R-module with Rad(M)E M. Then every submodule of a

w-supplemented module is w-supplemented.

Proof. Suppose that M is w-supplemented. Let N be a submodule of M and K be a

semisimple submodule of N. By Lemma 2.2.8, K is direct summand in M, that is,

M = K⊕ L for some submodule L of M. By modularity, N = K ⊕ (N ∩ L). Therefore N is w-supplemented.

Example 2.3.5. Artinian, semisimple and linearly compact modules are totally

Proposition 2.3.6. (by Faith (1973, p.356, 7.32A)) A ring R is a left V -ring in case the

following equivalent conditions are satisfied:

(1) Each simple left R-module is injective.

(2) Each left ideal is the intersection of maximal left ideals. (3) For any left R-module M, Rad(M) = 0.

Now we determine the relationship between w-supplemented and totally

w-supplemented modules.

Proposition 2.3.7. For a left V -ring R and an R-module M, the following statements

are equivalent:

(1) M is w-supplemented; (2) M is amply w-supplemented; (3) M is totally w-supplemented.

Proof. (1)⇔ (2) The proof is followed from Proposition 2.1.23

(1)⇔ (3) Let M be totally w-supplemented. Since M ⊆ M, M is also w-supplemented. Conversely, suppose M is w-supplemented and N be a submodule of M. We will show that N is w-supplemented. Let K be a semisimple submodule of N. Also K is semisimple submodule in M. If M is w-supplemented, then there exists a supplement

L in M such that M = K + L and K∩ L ≪ L. So K ∩ L ⊆ Rad(L) ⊆ Rad(M). Since R

is a left V -ring, then Rad(M) = 0. Thus K∩ L = 0, that is, M = K ⊕ L. By modularity, we have N = N∩ M = N ∩ (K ⊕ L) = K ⊕ (N ∩ L). So K has a supplement (N ∩ L) in

N. Hence N is w-supplemented.

Proposition 2.3.8. Let R be a Dedekind domain and M be a torsion R-module. Then

M is totally w-supplemented.

N

Rad(N)is semisimple. By Proposition 2.1.22, N is w-supplemented. Thus M is totally w-supplemented.

Lemma 2.3.9. Let R be a semilocal ring. Then every R-module is totally w-supplemented.

Proof. Let M be an R-module and N be a submodule of M. Since R is semilocal, then N

Rad(N) is semisimple. By Proposition 2.1.22, N is w-supplemented.

Theorem 2.3.10. Let a module M = M1⊕ M2 be a direct sum of submodules M1, M2

such that M2 is semisimple. Then M is totally w-supplemented if and only if M1 is

totally w-supplemented.

Proof. (⇒) Clear.

(⇐) Let N be a submodule of M. Since M2 is semisimple, then N∩ M2 is direct

summand in M2, that is, M2= (N∩ M2)⊕ L for some submodule L of M2. Thus

M = M1⊕M2= M1⊕(N ∩M2)⊕L. By modular law, N = (N ∩M2)⊕(N ∩(M1⊕L)).

Since N∩ (M1⊕ L) ∩ L = N ∩ L = 0, N ∩ (M1⊕ L) embeds in M1. Since M1is totally

w-supplemented, then N∩ (M1⊕ L) is w-supplemented. Thus N ∩ M2 is semisimple

because M2 is semisimple. It follows that N∩ M2 is w-supplemented. Therefore N is

w-supplemented by Proposition 2.1.12.

Lemma 2.3.11. Let R be a V -ring and M be an R-module. A module M is totally

w-supplemented if and only if M

L is totally w-supplemented for a simple submodule of M.

Proof. (⇐) Let N be a submodule of M. Then N + L

L is a submodule of M L. Suppose N + L L = M

L. So M = N + L. Since L is simple, either N∩ L = 0 or N ∩ L = L. If N∩ L = 0, then M = N ⊕ L. By assumption, M

L ∼= N is w-supplemented. If N∩ L = L,

then L⊆ N and N

submodule in N by Proposition 2.3.6. By Lemma 1.6.2, L is a direct summand in

N, that is, N = L⊕ L′ for some submodule L′ of N. Since N

L is w-supplemented, so is L′∼= N L. By Proposition 2.1.12, N = L⊕ L ′ _{is w-supplemented. Let} N + L L be a proper submodule of M L. Since M L is totally w-supplemented, N + L L ∼= N N∩ L is w-supplemented. Since L is simple, then either N∩ L = 0 or N ∩ L = L. If N ∩ L = 0,

then N

N∩ L ∼= N is w-supplemented. If N∩ L = L, then L ⊆ N. As we did in above, N

is w-supplemented.

Proposition 2.3.12. Let a module M = M1⊕ M2with Rad(M)E M where M1and M2

are totally w-supplemented modules. Then M is totally w-supplemented.

Proof. Let N be a submodule of M and K be a semisimple submodule of N. Since M1

and M2 are totally w-supplemented, M1and M2 are also w-supplemented. As a finite

direct sum of w-supplemented is w-supplemented, so is M. Then K has a supplement

L in M. That is, M = K + L and K∩ L ≪ L. By Corollary 2.1.6, M = K′⊕ L for

some submodule K′ of K. From Lemma 2.2.7, K = K′. Thus M = K⊕ L. So by modular law, N = K⊕ (N ∩ L), that is, K has a supplement N ∩ L in N. Hence N is

W-COATOMIC MODULES

3.1 W-Coatomic Modules

The notion of coatomic modules was introduced by Z¨oschinger (1980). G¨ung¨oro˘glu (1998) investigated some properties of coatomic modules and relationship between some classes of modules. The structure of coatomic modules over Dedekind domains are studied by G¨ung¨oro˘glu & Harmancı (1999). In this section, a characterization of w-coatomic modules are given and relationships between w-supplemented and

w-coatomic modules are investigated.

Definition 3.1.1. A module M with Soc(M) ̸= 0 is said to be w-coatomic, if every non-zero proper semisimple submodule of M is contained in a maximal submodule of

M.

Proposition 3.1.2. The following statements are equivalent for a module M: (1) M is w-coatomic module;

(2) For every semisimple submodule U of M, Rad(M

U) = M

U implies M U = 0.

Proof. (1) ⇒ (2) Let M be a w-coatomic and let Rad(M U) =

M

U for a semisimple

submodule U of M. Suppose M

U ̸= 0. So U is a proper submodule of M. But,

since Rad(M

U) = M

U, there is no maximal submodule of M containing U , this is a

contradiction.

(2)⇒ (1) Suppose that M is not w-coatomic. Let U be a proper semisimple submodule

of M. Then U is not contained in a maximal submodule of M. Thus Rad(M U) = M U. By (2), M U = 0, contradiction.

Obviously, any coatomic module which has nonzero socle is w-coatomic but converse is not true:

Example 3.1.3. Let Z and Q be the sets of integers and rational integers, respectively. Let us consider the Z-module M = Z

8Z⊕ Q. Then Soc(M) is nonzero, because Soc( Z

8Z) ∼= Z

2Z and Soc(Q) = 0. So, every proper semisimple submodule of M is of the form K⊕ 0 where K is a proper semisimple submodule of Z

8Z. Note that K ∼= Z

2Z and K is contained in the maximal submodule < 2 + 8Z > of Z 8Z. So,

K⊕ 0 is contained in the maximal submodule < 2 + 8Z > ⊕Q. We claim that M is

not coatomic. Because, otherwise by Lemma 1.9.9, Q must be coatomic. Since Q has no maximal submodule, that is,Q is not coatomic, a contradiction. Thus M is not coatomic.

It is proved that any finite direct sum of coatomic modules is coatomic in (G¨ung¨oro˘glu (1998), Corollary 5). We show that more of this property holds for

w-coatomic modules.

Proposition 3.1.4. For an R-module M, let M = M1+ M2. If M1 and M2 are

w-coatomic modules, then M is w-coatomic module.

Proof. Let U be a proper semisimple submodule of M. Let us consider U∩ M1. If

U∩ M1= M1, then M1 ⊆ U and so M1 is semisimple. If U∩ M2= M2, then M2 is

semisimple. Thus M = M1+M2is semisimple and so M is w-coatomic. In case M1⊆U

and U∩ M2̸= M2, since M2 is w-coatomic, there exists a maximal submodule K2 of

M2 such that U∩ M2⊆ K2. Clearly, M1+ K2 is maximal submodule of M1+ M2 and

then M1has a maximal submodule K1such that U∩M1⊆ K1. Thus K1+ M2is maximal

in M1+ M2, so U⊆ K1+ M2because parts of U that are included in M1are in K1and

parts of U staying in M2are in M2.

Corollary 3.1.5. Any finite sum of w-coatomic modules is w-coatomic. Lemma 3.1.6. Let

0→ L → M → N → 0

be a short exact sequence of R-modules. If both L and N are w-coatomic, then M is w-coatomic.

Proof. Let U be a proper semisimple submodule of M. Let us consider N as M L. Then U + L L is a semisimple submodule of M L. Suppose that U + L L = M L, then M = U + L. Since every semisimple module and L are w-coatomic, by Proposition 3.1.4 M is w-coatomic. Let U + L L be a proper submodule of M L. Since M L is w-coatomic, there

exists a maximal submodule K

L in M L such that U + L L ⊆ K

L for some submodule K of M containing L. Thus, K is maximal submodule in M containing U .

Lemma 3.1.7. For some submodule N of a module M, if M

N is w-coatomic which is not semisimple, then M is w-coatomic.

Proof. Let U be a proper semisimple submodule of M. Then U + N

N is a semisimple submodule of M N. If U + N N = M N, then M

N is semisimple, so we get a contradiction

because M

N is supposed to be not semisimple. Therefore

U + N N is proper semisimple submodule of M N. Since M N is w-coatomic, U + N

N is contained in maximal submodule

of M

N, say K

N. Thus, by third isomorphism theorem,

(M N) (K N) ∼ = M

K is simple, that is, K is

maximal in M containing U . Hence M is w-coatomic.

Example 3.1.8. Let Z be the set of integers. Let M = Z ⊕ Z_(p∞₎ where Z_(p∞₎ is the Pr¨ufer p-group for any prime p. Since Soc(Z) = 0 and Soc(Z_(p∞₎) ∼= Z

pZ, then Soc(M)̸= 0. Thus M is w-coatomic, because pZ ⊕ Z_(p∞₎ is maximal in M and every proper semisimple submodule of M is contained in pZ ⊕ Z_(p∞₎. But M

Z ∼=Z(p∞) is not

w-coatomic, sinceZ(p∞) has no maximal submodule.

Proposition 3.1.9. Let R be a left V -ring and let M be a w-coatomic R-module. Then

M

N is w-coatomic for a simple submodule N of M.

Proof. Let L

N be a semisimple submodule of M

N and let Rad( M L) = M L. Obviously, Soc(L N) = L

N. Since R is a left V -ring and N is a simple submodule of M, then N is

injective submodule in M and so it is direct summand of L. By (Kasch (1982), §9),

Soc(L N) =

Soc(L) + N

N and, it follows that L N =

Soc(L) + N

N . Thus L = Soc(L) + N.

Then L is semisimple submodule in M because N is simple. Since M is w-coatomic,

M = L.

Proposition 3.1.10. Let M be an R-module, let U be a proper semisimple submodule

of M and let V be a supplement of U in M. Then M is w-coatomic if and only if V is w-coatomic.

Proof. (⇒) Let U be a proper semisimple submodule of M. If V be a supplement of U , then M = U + V and U∩V ≪ V. Since U is semisimple in M, U = (U ∩V) ⊕U′

for some submodule U′ of U . Then it follows that M = U + V = (U∩V) +U′+ V . Therefore M = U′+ V . Since 0 = (U∩ V) ∩ U′ = V ∩ U′, then M = U′⊕ V. Let

Rad(V V′) =

V

V′ for a semisimple submodule V

′ _{of V . Then} M U′⊕V′ = U′⊕V U′⊕V′ ∼= V V′

is a radical module, that is, Rad( M

U′⊕V′) = M

U′⊕V′. Since U

′_{and V}′_{are semisimple}

submodules of M, so is U′⊕V′. By assumption, M

U′⊕V′= 0, that is, M = U

′_⊕V′_{. By}

minimality of V , V = V′. Hence V is w-coatomic.

is semisimple submodule in M and V is w-coatomic, by Proposition 3.1.2, M is

w-coatomic.

Lemma 3.1.11. Let M be an R-module with Rad(M) is w-coatomic. If M is w-supplemented, then M is w-coatomic.

Proof. Let U be a proper semisimple submodule of M and let Rad(M U) =

M

U. Since M

is w-supplemented, U has a supplement V in M, that is, M = U + V and U∩V ≪ V. Since U is semisimple in M, then U = (U∩V) ⊕U′ for some submodule U′ of U . It follows that M = U + V = (U∩V) +U′+ V and thus M = U′+ V . Because 0 = (U∩

V )∩U′= V ∩U′, then M = U′⊕V. By Lemma 1.7.2, Rad(M

U) = Rad(M) +U U . So Rad(M) +U U = M

U. It implies M = Rad(M)+U . Since Rad(M) and U are w-coatomic, M is w-coatomic by Proposition 3.1.4.

The proof of the following Corollary follows from Lemma 3.1.6 and Proposition 3.1.10.

Corollary 3.1.12. Let M be an R-module, let U be a proper semisimple submodule of

M and let V be a supplement of U in M. If M

U is w-coatomic, then V is w-coatomic.

Proposition 3.1.13. Let M be a reduced and w-supplemented module. Then M is

w-coatomic.

Proof. Let N be a proper semisimple submodule of M and let Rad(M N) =

M

N. Since M is w-supplemented, there exists a supplement K of N in M such that M = N + K

and N∩ K ≪ K. From Corollary 2.1.6, M = N′⊕ K for some submodule N′ of N because N is semisimple. By Lemma 1.3.2, Rad(M) = Rad(K)≤ K. By Lemma 1.7.2,

Rad(M N) = Rad(M) + N N , then M N = Rad(M) + N

N . It follows that M = Rad(M) + N.

By minimality of K, K = Rad(M) = Rad(K). Since M is reduced, K = 0. Thus M =

Proposition 3.1.14. For an R-module M let Rad(M)≪ M. If M is w-supplemented,

then M is w-coatomic.

Proof. Let N be a proper semisimple submodule of M and let Rad(M N) =

M N. By

assumption, N has a supplement in M. Therefore Rad(M

N) = Rad(M) + N N by Lemma 1.7.2. It implies M N = Rad(M) + N

N , that is, M = Rad(M) + N. Then M = N because Rad(M)≪ M.

We also have the following Corollary by Proposition 2.1.22 and Proposition 3.1.13. Corollary 3.1.15. Let R be a discrete valuation ring and M be a reduced R-module.

Then M is w-coatomic.

Lemma 3.1.16. Let R be a Dedekind domain and Rad(M)E M for an R-module M.

Then M

Rad(M) is w-coatomic.

Proof. Let R be a Dedekind domain and Rad(M)E M. Then M

Rad(M) is semisimple.

Since every semisimple module is w-coatomic, then M

Rad(M) is w-coatomic.

G¨ung¨oroˇglu (1998) has proved that every submodule of a coatomic module is coatomic over a discrete valuation ring. We proved the same Lemma under weaker condition.

Theorem 3.1.17. (by G¨ung¨oro˘glu (1998, Theorem 13)) Let M be a supplemented

module with Rad(M) is small in M. Then M

Rad(M) is semisimple and every submodule of Rad(M) is coatomic if and only if M is weakly supplemented and every submodule of M is coatomic.

Lemma 3.1.18. Let R be a discrete valuation ring and M be an R-module. Every

submodule of Rad(M) is w-coatomic if and only if every submodule U of M is w-coatomic.

Proof. (⇒) Let U be a submodule of M and N be a semisimple submodule of U. Let Rad(U

N) = U

N. We have by the isomorphism theorem and the modular law: U + Rad(M) N + Rad(M) = U + (N + Rad(M)) N + Rad(M) ∼= U U∩ (N + Rad(M))∼= U N + (U∩ Rad(M))

We claim thatU + Rad(M)

N + Rad(M) does not have a maximal submodule. Suppose to contrary

that U + Rad(M)

N + Rad(M) has a maximal submodule. By above isomorphisms, there is a

maximal submodule K of U containing N + (U∩ Rad(M)). Then K

N + (U∩ Rad(M)) is maximal submodule of U N + (U∩ Rad(M)). So K N is maximal submodule in U N.

This is impossible since Rad(U

N) = U

N, a contradiction. Thus, for U1= U + Rad(M)

and N1= N + Rad(M), we have Rad(

U1

N1

) = U1

N1

. Since R is a discrete valuation ring,

then M

Rad(M) is semisimple. So N1

Rad(M) is a direct summand in M

Rad(M), that is,

for submodule K1 of M M Rad(M) = N1 Rad(M) ⊕ K1

Rad(M). Then M = N1+ K1 and N1∩ K1= Rad(M). Since U1 N1 = Rad(U1 N1 )⊆ Rad(M N1 ) ∼= Rad( K1 Rad(M))⊆ Rad( M Rad(M)) = 0

then U1= N1. Hence U +Rad(M) = N +Rad(M) and so U = N +(U∩Rad(M)). From

Rad(U N) = U N = N + (U∩ Rad(M)) N ∼= U∩ Rad(M)

N∩ Rad(M), it follows that

U∩ Rad(M) N∩ Rad(M) = Rad(U∩ Rad(M)

N∩ Rad(M)). Since U∩ Rad(M) is submodule of Rad(M), then U ∩ Rad(M) is w-coatomic by assumption. Since N∩Rad(M) is submodule of N, then N ∩Rad(M) is

semisimple submodule of U∩ Rad(M). Therefore U ∩ Rad(M) = N ∩ Rad(M). Then

U = N. Thus U is w-coatomic.

(⇐) Let U be a submodule of Rad(M). So U is also a submodule of M. By assumption,

U is w-coatomic.

Corollary 3.1.19. Let M be a totally w-supplemented module. If Rad(M)≪ M, M is

Proof. If M is totally w-supplemented module, then M is also w-supplemented module.

By proposition 3.1.14, M is w-coatomic.

B¨uy¨ukas¸ık & Lomp (2008) defined w-local modules as follows:

A module M is called w-local if it has a unique maximal submodule. It is clear that a module is w-local if and only if its radical is maximal.

It is given an example in the following in order to show that any w-local module is not

w-coatomic and vice versa.

Example 3.1.20. Let M =Q ⊕ Z

pZ be an abelian group for any prime p. So J =

Q ⊕ 0 is the unique maximal submodule of M. Thus M is w-local. K = 0 ⊕ Z

pZ is

a proper semisimple submodule of M and since K is not contained in J, hence M is not w-coatomic. Conversely, let us consider the abelian group M =Z ⊕ Z_(p∞) for any

prime p. By Example 3.1.8, M is w-coatomic. qZ ⊕ Z_(p∞₎is a maximal submodule in

M for prime q. But qZ ⊕ Z_(p∞₎ is not unique in M. So M is not w-localZ-module.

We need an extra property to say the relationship between w-local and w-coatomic modules.

Proposition 3.1.21. For an R-module M, if M is a w-local and reduced module, then

M is w-coatomic.

Proof. Let N be a proper semisimple submodule of M. Because M is w-local, then its

radical is maximal. Therefore M

Rad(M) is semisimple. By Proposition 2.1.22, M is w-supplemented. Then by Proposition 3.1.13, M is w-coatomic, because M is reduced.

Proposition 3.1.22. For an R-module M, let M be a w-local module. If Rad(M)≪ M,

Proof. If M is w-local, then M

Rad(M) is semisimple. From Proposition 2.1.22 and