Multiplication modules which are distributive

Erdoğdu, V.

doi:10.1016/0022-4049(88)90031-x

Cited by 15 publications

(5 citation statements)

References 2 publications

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“…many authors have been interested in m~iltiplication modules and rings ( cf. 171, [8], [16], [20] and their references ). Hereditary rings, von Neumann regular rings and special primary rings ( SPIR's ) are some examples of this important class of rings.…”

Section: Introductionmentioning

confidence: 98%

Multiplication rings and graded rings

Escoriza

Torrecillas

1999

Communications in Algebra

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Section: Introductionmentioning

confidence: 98%

Multiplication rings and graded rings

Escoriza

Torrecillas

1999

Communications in Algebra

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“…Moreover, if M is a finitely generated faithful multiplication R-module then η and θ are lattice isomorphisms (Corollary 2.17). An R-module M is called distributive if L( R M ) is a distributive lattice (see, for example, [8]). A ring R is called arithmetical if it is a distributive R-module.…”

Section: Introductionmentioning

confidence: 99%

Mappings between the lattices of saturated submodules with respect to a prime ideal

Moghimi

Noferesti

Hosseini

2021

Hacettepe Journal of Mathematics and Statistics

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Let S p ( R M ) be the lattice of all saturated submodules of an R-module M with respect to a prime ideal p of a commutative ring R. We examine the properties of the mappings η : S p ( R R) → S p ( R M ) defined by η(I) = S p (IM ) and θ : S p ( R M ) → S p ( R R) defined by θ(N ) = (N : M ), in particular considering when these mappings are lattice homomorphisms. It is proved that if M is a semisimple module or a projective module, then η is a lattice homomorphism. Also, if M is a faithful multiplication R-module, then η is a lattice epimorphism. In particular, if M is a finitely generated faithful multiplication R-module, then η is a lattice isomorphism and its inverse is θ. It is shown that if M is a distributive module over a semisimple ring R, then the lattice S p ( R M ) forms a Boolean algebra and η is a Boolean algebra homomorphism.

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“…This gives N = aM = (N : R M)M, where (N : R M) = {r ∈ R : rM ⊆ N} is the residual of N by M. The ring R is called a multiplication ring if any ideal of R is a multiplication R-module in the above sense [12]. These concepts have attracted the interest of several authors in the last two decades, and have led to more information on the structure of R-modules (see for example [2,3,7,20,21]). In [9], multiplication modules in the category of R-modules graded by a group G were introduced and some interesting results were obtained (see also [8]).…”

mentioning

confidence: 99%

Finitely Generated Graded Multiplication Modules

Zamani

2011

Glasgow Math. J.

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Abstract. Let R = ⊕ i∈‫ޚ‬ R i be a ‫−ޚ‬graded ring and M = ⊕ i∈‫ޚ‬ M i be a graded R-module. Providing some results on graded multiplication modules, some equivalent conditions for which a finitely generated graded R-module to be graded multiplication will be given. We define generalised graded multiplication module and determine some of its certain graded prime submodules. It will be shown that any graded submodule of a finitely generated generalised graded multiplication R-module M has a kind of primary decomposition. Using this, we give a characterisation of graded primary submodules of M. These lead to a kind of characterisation of finitely generated generalised graded multiplication modules.2010 Mathematics Subject Classification. 13A02, 13C05, 13C13, 13C99. Introduction.Let R be a commutative ring with a non-zero identity and M be a unital R-module. Then M is called a multiplication module if any submodule N of M has the form aM for some ideal a of R [4]. This gives N = aM = (N : R M)M, where (N : R M) = {r ∈ R : rM ⊆ N} is the residual of N by M. The ring R is called a multiplication ring if any ideal of R is a multiplication R-module in the above sense [12]. These concepts have attracted the interest of several authors in the last two decades, and have led to more information on the structure of R-modules (see for example [2, 3, 7, 20, 21]). In [9], multiplication modules in the category of R-modules graded by a group G were introduced and some interesting results were obtained (see also [8]). Nevertheless, based on the author's knowledge few results are known on this concept. In this paper we deal with the category of graded multiplication modules over a ‫-ޚ‬graded ring R and obtain some concerning results. More precisely, among other things, it will be shown that for a graded submodule N of a finitely generated generalised graded multiplication R-module M, there exists a primary decomposition N = p∈P N(p), where P denotes the set of all minimal primes of (N : R M). We also show that for a finitely generated graded multiplication R-module M, this kind of primary decomposition for each submodule N of M, together with other two conditions, gives that M must be a generalised graded multiplication R-module. It should be noted that all of the results remain valid if one replaces the grading group ‫ޚ‬ with any torsion-free abelian group G having a total order compatible with its structure. However, for the convenience we are interested in more concrete abelian group ‫.ޚ‬ The organisation of the paper is as follows. In Section 2, we collect some preliminary results on graded multiplication modules. Some characterisations for

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Multiplication modules which are distributive

Cited by 15 publications

References 2 publications

Multiplication rings and graded rings

Multiplication rings and graded rings

Mappings between the lattices of saturated submodules with respect to a prime ideal

Finitely Generated Graded Multiplication Modules

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