On the prime submodules of multiplication modules

Ameri, R.

doi:10.1155/s0161171203202180

Cited by 52 publications

(50 citation statements)

References 5 publications

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“…Trace (M, JM ) = JM . Hence we can describe the -product of submodules of multiplication modules: We see that the product of submodules of multiplication modules over commutative rings as defined in [1] coincides with our -product. With our approach it is not necessary to show that this product is independent of the choice of representing ideals I and J for the submodules N and L.…”

Section: All Elements Ofmentioning

confidence: 89%

Prime Elements in Partially Ordered Groupoids Applied to Modules and Hopf Algebra Actions

Lomp

2005

J. Algebra Appl.

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Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M ) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M ) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.

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Section: All Elements Ofmentioning

confidence: 89%

Prime Elements in Partially Ordered Groupoids Applied to Modules and Hopf Algebra Actions

Lomp

2005

J. Algebra Appl.

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“…For the special case, that is multiplication module over a commutative ring, the definition between two submodules is well defined. (see for example paper of Ameri [1] or Azizi [2]). …”

Section: Proposition 46mentioning

confidence: 99%

“…Hence the observation of further properties of prime submodules can be done in a multiplication module. However, most of the previous paper presented multiplication modules over a commutative rings with unit (for example Ameri [1], Azizi [2], El-Bast [5] and [6], Tekir [10]). By the commutativity one can analyze the properties of prime submodules in a simple way.…”

Section: Introductionmentioning

confidence: 99%

Endo-prime submodules in endo-multiplication modules

Wijayanti¹

2014

imf

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There are various approaches to transfer the prime ideal notion to prime submodule. The definition of primeness in modules have appeared in many ways. In this paper we define endo-prime submodules and show that this definition and prime submodules introduced by Sanh et al. [8] are coincide. Moreover, we also present the endo-prime radical and its properties. Then we apply our results to the endo-multiplication modules, i.e. a multiplication module over its ring endomorphisms.Mathematics Subject Classification: 16D50, 16D70, 16D80

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“…Various generalizations of prime (primary) ideals are studied in [1][2][3][4][5][6][7][8]. The class of prime submodules of modules as a generalization of the class of prime ideals has been studied by many authors; see, for example, [9,10]. Then many generalizations of prime submodules were studied such as weakly prime (primary) [11], almost prime (primary) [12], 2-absorbing [13], classical prime (primary) [14,15], and semiprime submodules [16].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, any proper idempotent submodule of is almost semiprime. If is a multiplication -module and = and = are two submodules of , then the product of and is defined as = ( )( ) = ( ) ; see [9]. In particular, one has…”

Section: Introductionmentioning

confidence: 99%