On finitely generated projective ideals in commutative rings

Naoum, Adil G.

doi:10.1007/bf01849617

Cited by 9 publications

(8 citation statements)

References 3 publications

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“…Now we give conditions under which a semi-multiplication module becomes projective (compare with the relation between multiplication modules and projective modules [7]). = (al,...,an).…”

Section: W 2 Basic Resultsmentioning

confidence: 99%

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Semi-multiplication modules

Hummadi

Naoum

1993

Period Math Hung

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“…Now we give conditions under which a semi-multiplication module becomes projective (compare with the relation between multiplication modules and projective modules [7]). = (al,...,an).…”

Section: W 2 Basic Resultsmentioning

confidence: 99%

“…Now let X = (zl,...,zn) E U j', hence ~oi(~_~aizi) = 0, this implies that Cz t = 0. Consequently M is projective [4]. 9…”

Section: W 2 Basic Resultsmentioning

confidence: 99%

Semi-multiplication modules

Hummadi

Naoum

1993

Period Math Hung

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“…. ln), then it can be checked easily t h a t V t --M t V t and V ± ----ann (Mr); and it follows from [11], [6] t h a t P* is projective. Using the same argument, one can show t h a t P** is projective.…”

Section: ~ L(aj) ~ Li(at) X I = L(aj) 1 I a Sx I : L(aj) Lj(x)mentioning

confidence: 99%

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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“…Multiplication modules have received a lot of attention in recent years (see, for example, [1], [3]- [6], [8], [9]). W. W. Smith [9,Theorem 1] proved that any projective ideal of R is a multiplication R-module.…”

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“…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%

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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On finitely generated projective ideals in commutative rings

Cited by 9 publications

References 3 publications

Semi-multiplication modules

Semi-multiplication modules

On finitely generated projective modules and multiplication modules

Multiplication modules and projective modules

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