EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 410-415 ISSN 1307-5543 – ejpam.com Published by New York Business Global Direct summand of serial modules AL-Housseynou BA1,∗, Mankagna Albert Diompy1, André Souleye Diabang2 1 Département de Mathématiques et Informatique, Faculté des Sciences et Techniques, Université Cheikh Anta DIOP, Dakar, Sénégal 2 Département de Mathématiques , UFR Sciences et Technologies, Université de Thiès, Thiès, Sénégal Abstract. Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini ”Module Theory” which states that: is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules. 2020 Mathematics Subject Classifications: 13C60, 13C05, 13C13 Key Words and Phrases: Uniserial, serial, local, direct summand 1. Introduction Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. The target of this paper comes from the following state- ment. Is any direct summand of serial module serial? This is an open problem listed in the Module theory book of Alberto Facchini. Some results has been obtained if the base ring is commutative or noetherian ... Other results are obtained in this paper for particular rings and modules. A module M is said to be a prime module if for every submodule N of M , Ann(M) = Ann(N). A module is M is said to be faithful if Ann(M) = 0. A module M is said to be finitely cogenerated if its socle is essential in N and finitely generated Lemma 1: Let M be an uniserial module over a ring R. M is said to be of type 1 if at least one of the following hold. (1) M is projective; (2) M is injective; (3) M is artinian; ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4973 Email addresses: alhousseynou.ba@ucad.edu.sn (Al-H BA ), albertdiompy@yahoo.fr (M. A. Diompy), andrediabang@yahoo.fr (A. S. Diabang ) https://www.ejpam.com 410 © 2024 EJPAM All rights reserved. Al-H. BA, M. A. Diompy, A. S. Diabang / Eur. J. Pure Appl. Math, 17 (1) (2024), 410-415 411 (4) M is noetherian; (5) R is commutative; (6) R is a right noetherian. Proof: To see the proof refer from example 2.3 of [1]. Proposition 1: Let R be ring and M be a left local module over R, then End(M) is local. Proof. Let M be an hollow module. Hence M is finitely generated. Let f : R −→ M an homo- morphism which is an epimorphism. Therefore R/Ann(M) is isomorphic to M . Hence R/Ann(M) is hollow. It follows from theorem 4.1 of [5] that End(R/Ann(M)) is local. Thus End(M) is local. Theorem 1: Let M1, ...,Mn be uniserial local modules. Let M = ⊕ i∈I Mi a serial module. Then every summand of M is serial. Proof. It results from proposition 1 that End(Mi) is local. Therefore every direct summand of M is serial. Proposition 2: Let R be ring and M a finitely cogenerated, prime and faithful R- module. Then R as a left R-module is uniserial. Moreover End(R) is a local ring. Proof. Let M be a finitely cogenerated module over R. It is well known that any finitely cogen- erated module has a small submodule. Let K be its small submodule. Then f : R −→ K is an epimorphism. f : R −→ K ↓ ↙ R/Ann(K) By the first isomorphism theorem, R/Ann(K) is isomorphic to K. As M is a prime and faithful module then Ann(K) = Ann(M) = 0. Therefore R is simple as a left R-module. Hence R is uniserial because {0} ⊂ R. Since R is simple then for ever endomorphism of R is an automorphism. Hence for every endomorphism g : R −→ R there exists always a endomorphism g : R −→ R such that g ◦ h = Id and h ◦ g = Id. That implies End(R) is a division ring. It is well know that any division ring is a local ring. Corollary 1: Let R = ⊕ i∈I Ri where (Ri)i∈I is family of rings such that there exists a finitely cogen- erated, prime and faithful Ri0-module with i0 ∈ I. Then the following conditions are verified: Al-H. BA, M. A. Diompy, A. S. Diabang / Eur. J. Pure Appl. Math, 17 (1) (2024), 410-415 412 (1) Each Ri is uniserial as a left Ri-module for every i ∈ I. (2) Every summand of R is serial. Proof. (1) Let M be a left finitely cogenerated prime and faithful Ri0-module. Then M is a module over every Ri with i ∈ I by the following homomorphism: f : Ri −→ Ri0 ×M −→ M r 7−→ (f(r),m) 7−→ f(r)m It follows from proposition 2 that Ri is simple hence uniserial for every i ∈ I. (2) It results from proposition 2 that End(Ri) is a local ring for every i ∈ I. Thus every summand of R is serial. In the following corollary we show that a finitely generated module M is serial under certain conditions and every summand of M is serial. Corollary 2: Let R be a ring and M = ⊕n i=1Mi a finitely generated and prime module Such that M has a small submodule. Then M is serial and so is every summand of M . Proof. Let M a finitely generated prime module. Let f : R −→ K be an epimorphism where K is a small submodule. f : R −→ K ↓ ↙ R/Ann(K) By the first isomorphic theorem R/Ann(K) ≃ K. Let g : R −→ Mi another epimorphism for every 1 ≤ i ≤ n f : R −→ Mi ↓ ↙ R/Ann(Mi) We have also R/Ann(Mi) ≃ Mi. Since M is a prime module Ann(K) = Ann(Mi), there- fore R/Ann(K) = R/Ann(Mi) ≃ Mi is simple for 1 ≤ i ≤ n. Therefore M = ⊕n i=1Mi is semisimple. Proposition 3: Let R be a ring and M an uniserial R-module, then End(M) is local if, (1) M is self-projective; (2) M is self-injective; (3) M is a free module. Proof : (1) As M is uniserial, hence it is uniform and indecomposable. Let f ∈ End(M) therefore ker f ∩ ker(1 − f) = 0. Since M is uniform, then f or 1 − f is a monomorphism. If M Al-H. BA, M. A. Diompy, A. S. Diabang / Eur. J. Pure Appl. Math, 17 (1) (2024), 410-415 413 is self-projective then any sequence 0 −→ N −→ M −→ M −→ 0 is split. Hence an endomorphism of M is surjective. Therefore End(M) is local. (2) Let f ∈ End(M) therefore ker f ∩ ker(1 − f) = 0. Then f is injective. Since M is self-injective, then f(M) is a direct summand of M that is M = f(M) ⊕ N . But M is uniserial, hence M is indecomposable. Therefore f(M) = M which states that f is an epimorphism. Thus f is an automorphism. End(M) is a division ring (3) Since any free module is projective then End(M) is local by lemma 1. Theorem 2: Let R be a ring and M = ⊕n i=1Mi a serial module with Mi self- projective(resp. self-injective or free). Then any direct summand of M is serial. Proof. Let R be a ring and M = ⊕n i=1Mi a serial module with Mi self-projective(resp. self- injective or free. It results from proposition 3 that the endomorphism ring of any self- projective(resp. self-injective or free) module is local. It follows from proposition 2.2 of [4] that the direct summand of any direct sum of uniserial modules with local endomorphism rings is serial. Proposition 4: Let R be a ring. (1) If R is semisimple ring, then every a left module M over R is serial. Moreover, every direct summand of M is serial. (2) If R is principal ring and M a left serial R-module, then every direct summand of M is serial. Proof: (1) Let M be a left R-module. Since R is semisimple hence, M is semisimple. Let M = ⊕ i∈I Mi with Mi simple. It is well know that any simple module is uniserial. Hence, M is serial. Let S = End(Mi) an endomorphism ring of simple module Mi, by Schur’s Lemma S is a division ring. Therefore S is a local ring. Thus every direct summand of M is serial. (2) If R is principal ring, then every ideal over R is principal is cyclic( finitely generated). Thus by the definition of noetherian ring, R is noetherian. Theorem 3: Let R be a semisimple or principal ring and M = ⊕n i=1Mi a serial module. Then every direct summand of M is serial. Proof: Let R be a semisimple M = ⊕n i=1Mi a serial module. It results from proposition 4 that End(Mi) is a local endomorphisme ring for any 1 ≤ i ≤ n. By the proposition 2.2 of [4] that any direct summand of M is a direct sum of serial module. REFERENCES 414 Assume R is a principal ring. It results from proposition 4 that R is a noetherian ring. It follows from example 2.3 of [1] that any direct summand of M is serial. Proposition 5: Let R = ⊕n i=1Ri be a serial semiperfect ring then (1) End(Ri) is local, (2) every direct summand of R is serial. Proof: As R is semiperfect then R is a sum of is local ring. Hence Ri is local for any 1 ≤ i ≤ n. It results from proposition 1, that End(Ri) is local. Therefore any direct summand of R is serial. Theorem 4: Let R be a local ring and M = ⊕n i=1Mi a finitely generated serial module. Then every direct summand of M is serial. Proof. Let R be a local ring and M = ⊕n i=1Mi a finitely generated serial module left R-module. Assume Mi a cyclic module. By the following diagram, f : R −→ Mi ↓ ↙ R/Ann(Mi) R/Ann(Mi) is isomorphic to Mi. As R is local, it has a unique maximal ideal, J . Let Ī be an ideal of R/Ann(Mi) then Ī = I/Ann(Mi) with I an ideal of R and Ann(Mi) ⊆ J . Therefore J/Ann(Mi) is the unique maximal ideal of R/Ann(Mi). Hence, R/Ann(Mi) is a local ring. Thus Mi is local. References [1] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. 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