On the ring of endomorphisms of a finitely generated multiplication module

Naoum, Adil G.

doi:10.1007/bf02651092

Cited by 14 publications

(13 citation statements)

References 7 publications

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“…If A is finitely generated, then Hom(A,A) is finitely generated ( [7], Proposition 2.1). But F(A) = C"(A) by C orollary 2.2 and Co(A) = Hom(A,A), therefore F(A) is finitely generated.…”

Section: The Trace Of a Module A In A Module Bmentioning

confidence: 98%

On the module of homomorphisms into projective modules and multiplication modules

Naoum

Kider

1996

Period Math Hung

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“…If A is finitely generated, then Hom(A,A) is finitely generated ( [7], Proposition 2.1). But F(A) = C"(A) by C orollary 2.2 and Co(A) = Hom(A,A), therefore F(A) is finitely generated.…”

Section: The Trace Of a Module A In A Module Bmentioning

confidence: 98%

On the module of homomorphisms into projective modules and multiplication modules

Naoum

Kider

1996

Period Math Hung

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“…Naoum's theorem [5,Theorem 4.1] gives necessary and sufficient conditions for a finitely generated multiplication module to be projective. It is perhaps tempting to think that there is an analogue to Corollary B giving necessary and sufficient conditions for a general multiplication module to be projective.…”

Section: Proof the Necessity Is Proved In Corollary 8 Conversely Smentioning

confidence: 99%

“…Moreover, it can be seen from the proof of Theorem A that the theorem remains true if "finitely projective" is replaced by "cyclicly projective". We chose to prove Theorem A in the form given to bring out its relationship with Naoum'swork in [5].…”

Section: Proof the Necessity Is Proved In Corollary 8 Conversely Smentioning

confidence: 99%

“…On the other hand, it is well known that a free module is a multiplication module if and only if it has rank 1 (see, for example, [3, Lemma 2.1]). Naoum [5,Theorem 4.1] proved that a finitely generated module M is a multiplication module whose annihilator is generated by an idempotent in R if and only if M is a projective module whose endomorphism ring is commutative (see also [6,Theorem 2.1] and [8,Theorem 11]). In this note we investigate further the relationship between multiplication modules and projective modules.…”

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confidence: 99%

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Multiplication modules and projective modules

Smith

1994

Period Math Hung

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In this note all rings are commutative and all modules are unital. Let R be a ring. An R-module M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM. A submodule N of a general R-module M will be called fully invariant if ~(N) C N for every endomorphism ~ of M. The R-module M will be called finitely projective if, for every finitely generated submodule N of M there exist a positive integer n, elements mi E M (1 < i < n) and R-homomorphisms: 6i : M --, R (1< i < n) such thatfor all x in N. Note that every projective module is finitely projective, and every finitely generated finitely projective module is projective, by the Dual Basis Lemma (see, for example, [2, p. 203]). Our purpose is to prove the following result which generalizes [5, Theorem 4.1]. THEOREM A. Let R be any ring and M an R-module with annihilator A and endomorphism ring A. Then the following statements are equivalent. (i) M is a multiplication module. (iX) M is a finitely projective (R/a)-module and every submodule of M is fully invariant (iii) M is a finitely projective (R/A)-module and h is a commutative ring. (iv) M is a finitely projective (R/A)-module and M is a locally cyclic Rmodule.( v) M is a finitely projective (R/A)-module and O(m)M C_ Rm for all m E M and every R-homomorphism 0 : M ~ R/A.With the above notation, every projective R-module M is a projective (R/A)-module and hence a finitely projective (R/A)-module by the Dual Basis Lemma. Thus the above theorem gives the following immediate corollaries, the second of which also generalizes [5, Theorem 4.1].ring.

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“…Let E(M) = S be the ring of endomorphisms of M. It is known that if M is a multiplication module, then E(M) is commutative [6], and if M is a finitely generated multiplication module, then E(M) is isomorphic to R/ann (M), [4]. In this paper we give further results about E(M) for M a multiplication module.…”

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confidence: 92%

On the ring of endomorphisms of a multiplication module

Naoum

1994

Period Math Hung

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Let R be a commutative ring with 1, and M is an R-module. Let E(M) = S be the ring of endomorphisms of M. It is known that if M is a multiplication module, then E(M) is commutative [6], and if M is a finitely generated multiplication module, then E(M) is isomorphic to R/ann (M), [4]. In this paper we give further results about E(M) for M a multiplication module.Let us call an endomorphism f of a module M a diagonal endomorphism if for all a E M, 3r E R, r depends on a, such that f(a) = ra. In Sec. 1 of this paper we show that every endomorphism f of a multiplication module is diagonal. In fact, we show that for each finite set al, a2,..., an E M, qr E R such that f(ai) = rai 1 < i < n, (see Theorem 1.5). On the other hand, we show that E(M) may not be isomorphic to R/ann (M) if M is not finitely generated. (See Example 1.) Motivated by the work in [9] and [3] on the ring of endomorphisms of a projective module, we study in Sec. 2 the Jacobson Radical of E(M). It turns out that J(E(M) has the same characterization for both projective modules and flat multiplication modules. Finally, we remark that for information on multiplication modules, see [1], [2], [5], [8]. w 1. The ring E(M) We start by the following: LEMMA 1.1. Let M be an R-module which is locally cyclic, and let f be an endomorphism of M. Then, for each a E M, 3ra E R ra depends on a, such that f(a) = raa.

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On the ring of endomorphisms of a finitely generated multiplication module

Cited by 14 publications

References 7 publications

On the module of homomorphisms into projective modules and multiplication modules

On the module of homomorphisms into projective modules and multiplication modules

Multiplication modules and projective modules

On the ring of endomorphisms of a multiplication module

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