The residual of finitely generated multiplication modules

Naoum, Adil G.; Hasan, Mokhtar M.

doi:10.1007/bf01194187

Cited by 27 publications

(8 citation statements)

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“…A characterization of finitely generated multiplication modules by matrices was given in [8]. Now we have the following characterization of semi-multiplication modules.…”

Section: W 2 Basic Resultsmentioning

confidence: 99%

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Semi-multiplication modules

Hummadi

Naoum

1993

Period Math Hung

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“…A characterization of finitely generated multiplication modules by matrices was given in [8]. Now we have the following characterization of semi-multiplication modules.…”

Section: W 2 Basic Resultsmentioning

confidence: 99%

“…We have the following Remark. [8]. 9 Finally, we give an example of a semi-multiplication module such that ann (M) + T(M) = R, but M is not projective.…”

Section: If M Is a Finitely Generated Projective Module With A Local mentioning

confidence: 99%

Semi-multiplication modules

Hummadi

Naoum

1993

Period Math Hung

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“…H e n c e if P is faithful, t h e n e ----0 a n d trace M ~ 1. Moreover, it follows f r o m Corollary 1.4 t h a t air~k -~ akrti for all 1 ~ i, j, k ~ n. A n d b y [10], T h e o r e m 1.1, P is a multiplication module. I f P is n o t faithful, t h e n let P1 ~ e R • P .…”

Section: Proof F I R S T We H a V E U -~ U M A N D U~: Ann (M) Momentioning

confidence: 94%

“…,an). There exists an n × n matrix M 1 ---- [rii ] such that U M 1 : U, t r a c e M 1~ 1, and airlk ----akrtl, V l ~ i, j, k ~ n (see [10], Theorem 1.1).…”

Section: Corollary 13 I [ P Is As Above Thenmentioning

confidence: 98%

On finitely generated projective modules and multiplication modules

1989

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IntroductionL e t R be a commutative ring with 1. If I is a finitely generated (f.g.) ideal in R, then I is a projective ideal iff I is a multiplication ideal and ann (I) is generated b y an idempotent [9]. The main goal of this paper is to s t u d y the relationship between f.g. projective modules and multiplication modules.Recall t h a t a right module P is said to be a multiplication module if every submodule N of P has the form P I for some ideal I of R [1]. On the other hand, a f.g. projective module P is called hereditarily projective if every homomorphic image of P into a f.g. projective module is projective. Equivalently, if l(P) is a projective ideal of R for each linear functional I on P (i.e. l E P*, the dual of P). The projective module P is called cohereditarily projective if the closure of each f.g. submodule N of P is a direct s u m m a n d in P [2].The main result of this paper is that for indecomposable f.g. modules P over P.P. rings R, P is hereditarily projective iff P is a multiplication module and ann (P) is generated b y an idempotent (see Theorem 2.7).I t was proved in [2] t h a t a f.g. module P is hereditarily projective iff P* is cohereditarily projective. I t is proved in this paper t h a t these two concepts are equivalent if P is indecomposable and R is a P.P. ring (see Theorem 2.8).Finally, we remark t h a t all rings in this paper are commutative with 1, and all modules are unitary. For basic ideas on multiplication ideals see [4]. §-1. Preliminaries

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“…Let be a submodule of and let be an ideal of ; the residual submodule of by is defined as ( : ) = { ∈ : ⊆ }. These two residual ideals and submodules were proved to be useful in studying many concepts of modules; see, for example, [18,19]. A proper submodule of an -module is a prime submodule if, whenever ∈ for ∈ and ∈ , ∈ or ∈ ( : ).…”

Section: Introductionmentioning

confidence: 99%