Tez Izleme 6

Ph.D. Thesis Report w - Coatomic Modules Tuˇgba G¨ uroˇglu June 17, 2009 Abstract In this work, we study w - coatomic modules and we try to investigate some properties of w - coatomic modules.

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Introduction

Throughout this note we assume that R is an associative ring with unity and all modules are unital left R- modules, unless otherwise mentioned. Let R be a ring and M be an R-module. Rad(M ) and Soc(M ) will denote Jacobson radical and socle of M , respectively. A module M is said to be semisimple , if every submodule of M is a direct summand in M . A module M is said to be coatomic if every proper submodule of M is contained in a maximal submodule of M (see [4]), equivalently, for a submodule N of M , whenever Rad(M/N ) = M/N , then M = N . Semisimple modules, finitely generated modules, hollow modules and local modules are coatomic modules. The submodule T (M ) = {m ∈ M : rm = 0 for some 0 6= r ∈ R} is called the torsion submodule of M and if M = T (M ) then M is called a torsion module. A module M is supplemented, if every submodule N of M has a supplement, i.e., a submodule K minimal with respect to N + K = M . K is a supplement of N if and only if N + K = M and N ∩ K ¿ K.

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Properties of w - Coatomic Modules

Definition 2.1. A module M is called w-coatomic if every proper semisimple submodule of M is contained in a maximal submodule of M . Proposition 2.2. The following statements are equivalent for a module M : (1) M is w - coatomic. (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 6= 0, so U is proper submodule of M . But since Rad(M/U ) = M/U , there is no maximal submodule of M containing U , 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. Example 2.3. Obviously any coatomic module is w-coatomic but the converse is not true. Let Z denote the ring of integers. Consider the Z-module M = ⊕N Mn where Mn = Z for all n ∈ N. Since the only semisimple submodule of M is 0 and is contained in maximal submodule ⊕pZ, M is w-coatomic. But by ([6], page 155), M = ⊕N Mn is not coatomic . Lemma 2.4. For some submodule N of M , if M/N is w - coatomic, then M is w - coatomic. Proof. Let U be a semisimple submodule of M . Then U + N/N is a semisimple submodule of M/N . Since M/N is w - coatomic, U +N/N is contained in a maximal submodule of M/N , say K/N . Thus M/N/K/N ∼ = M/K is simple, that is, K is maximal in M containing U . Hence M is w - coatomic. Example 2.5. Consider the Z-module N . Let M = N ⊕ Q. Then M is w - coatomic, but the factor module M/N ∼ = Q is not w - coatomic because Q has no maximal submodule. Thus the converse statement of lemma 2.4 is not true. Corollary 2.6. If M/N is semisimple for proper submodule N of M , then M is w - coatomic. Proof. Let M/N be a semisimple. Then M/N is coatomic and M/N is also w - coatomic and by lemma 2.4, M is w - coatomic. 2

Proposition 2.7. Let M = ⊕ni=1 Mi where every Mi is w - coatomic. Then M is w - coatomic. Proof. Let’s use the induction on n. If n = 1, then M is w - coatomic because M1 is w - coatomic. Suppose for n = k, M = ⊕ki=1 Mi is w - coatomic. Now let’s show for n = k + 1, M is w - coatomic. It can be written as k k k M = ⊕k+1 i=1 Mi = (⊕i=1 Mi ) ⊕ Mk+1 . Then M/Mk+1 = ⊕i=1 Mi . Since ⊕i=1 Mi is w - coatomic, by lemma 2.4, M is w - coatomic. Proposition 2.8. Let M be an R- module, U be a semisimple submodule of M and V be a supplement of U in M . Then M is w-coatomic if and only if V is w - coatomic. Proof. (⇒) Let M be a w-coatomic. If V is a supplement of U , then M = U + V and U ∩ V ¿ V . Since U is semisimple, U = (U ∩ V ) ⊕ U 0 for some submodule U 0 of U . It follows that M = U + V = (U ∩ V ) + U 0 + V and so M = U 0 + V . Because 0 = (U ∩ V ) ∩ U 0 = V ∩ U 0 , then M = U 0 ⊕ V . Let U = 0. Then M = V . By assumption, V is w-coatomic. Now suppose U 6= 0. Let Rad(V /V 0 ) = V /V 0 for a semisimple submodule V 0 of V . Then M/(U 0 ⊕ V 0 ) = (U 0 ⊕ V )/(U 0 ⊕ V ) ∼ = V /V 0 is radical module. Since U 0 and V 0 are semisimple submodules, by assumption, M/(U 0 ⊕ V 0 ) = 0, that is, M = U 0 ⊕ V 0 . Thus V = V 0 . (⇐) Let V be a w-coatomic and V be a supplement of U . Then M = U + V and U ∩ V ¿ V . For semisimple submodule U 0 of U , M = U 0 ⊕ V . Since M/V ∼ = U 0 is semisimple, by corollary 2.6, M is w-coatomic. Lemma 2.9. If every maximal submodule of M is direct summand, then M is w - coatomic. Proof. Let K be a maximal submodule of M . Then M = K ⊕ K 0 for some submodule K 0 of M . So M/K ∼ = K 0 is simple. Then K 0 is semisimple. Since every semisimple module is coatomic, K 0 is coatomic. Thus K 0 is w coatomic and by lemma 2.4, M is w - coatomic. Lemma 2.10. Let M be a w - coatomic module. Then M/Soc(M ) contains a maximal submodule. Proof. Suppose that M/Soc(M ) does not contain a maximal submodule. By assumption, Soc(M ) is contained in a maximal submodule K in M since Soc(M ) is semisimple. Then M/K is simple. It follows that M/Soc(M )/K/Soc(M ) is simple. So K/Soc(M ) is maximal submodule in M/Soc(M ), contradiction. Thus M/Soc(M ) contains a maximal submodule.

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Lemma 2.11. Let M be w-coatomic module. Then Rad(M ) 6= M . Proof. Suppose that Rad(M ) = M . Let N be a semisimple submodule of M . Since Rad(M ) = M , then there is no maximal submodule containing N , contradiction. Thus Rad(M ) 6= M . Example 2.12. Let M be a w-coatomic module. Then every submodule of M is not w-coatomic. For example; consider the submodule Rad(M ) of M . Rad(M ) would be a w-coatomic, then every semisimple submodule of Rad(M ) were contained in a maximal submodule in Rad(M ). But Rad(M ) has no maximal submodule. Hence Rad(M ) is not w-coatomic. Lemma 2.13. Let R be a DVR. Then every R-module M is w-coatomic. Proof. In DVR, M/Rad(M ) is semisimple. Then by corollary 2.6, M is wcoatomic. We call M a semilocal module if M/Rad(M ) is semisimple. Lemma 2.14. Every semilocal module is w - coatomic. Proof. Let M be a semilocal module. Then M/Rad(M ) is semisimple. By corollary 2.6, M is w - coatomic. Example 2.15. Consider the Z module Z where Z is the ring of integers. Z module Z is w - coatomic but Z is not semilocal module, that is, the converse of the above lemma is not true. Let M be a module and U, V be submodules of M . We say that V is a weak supplement of U in M if M = U + V and U ∩ V ¿ M . M is called weakly supplemented if every submodule of M has a weak supplement. Proposition 2.16. Every weakly supplemented module is w - coatomic. Proof. Let M be weakly supplemented module and let N be a semisimple submodule of M . Then N has a weak supplement K in M for some submodule K of M such that M = N + K and N ∩ K ¿ M . Since N is semisimple, N = (N ∩K)⊕N 0 for some submodule N 0 of N . It follows that M = N 0 ⊕K. So M/K ∼ = N 0 is semisimple, thus M/K is w - coatomic. By lemma 2.3, M is w - coatomic.

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Proposition 2.17. Let R be a hereditary ring. Then M is w - coatomic if and only if every nonzero injective submodule of M is w - coatomic. Proof. (⇒) Let M be a w - coatomic and N be a nonzero injective submodule of M . Let L be a semisimple submodule of N . Let L = 0. By assumption, L is contained in a maximal submodule K of M . Since N is injective submodule of M , then N + K/K is injective submodule of M/K and N + K/K is direct summand in M/K. That is, M/K = (N + K)/K ⊕ N 0 /K for some submodule N 0 of M . If N + K/K is not proper submodule of M/K, then N +K/K = M/K. Since N +K/K ∼ = N/N ∩K and M/K is simple, so N ∩K is maximal submodule in N containing L. Let N +K/K is proper submodule of M/K. Since M/K is simple, then N + K/K = 0 or N + K/K = M/K. If N + K/K = 0, then M/K = N 0 /K, contradiction. Thus N/N ∩ K ∼ = N + K/K = M/K, i.e., N ∩ K is maximal submodule in N containing L. Let L 6= 0. Similar to above, N ∩ K is maximal submodule in N containing L because L is submodule of N and K. (⇐) Let N be an injective submodule of M . Then M = N ⊕ K for some submodule K of M . Thus M/K ∼ = N is w - coatomic by assumption. By lemma 2.4, M is w - coatomic. Lemma 2.18. Let R be a Dedekind domain. If M is torsion module, then M is w - coatomic. Proof. Let M be a torsion module. By ([12], corollary 2.7), M/Rad(M ) is semisimple and by corollary 2.6, M is w - coatomic. Lemma 2.19. Let R be a Dedekind domain. Let M be a torsion module. Then every submodule of M is w - coatomic. Proof. Let M be a torsion module and N be submodule of M . Then N is torsion module. By lemma 2.18, N is w - coatomic. Lemma 2.20. Let R be a Dedekind domain and K be the field of quotients of R. Then R K is w - coatomic. Proof. By ([12], lemma 2.8), R K is weakly supplemented and by proposition 2.16, R K is w - coatomic.

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