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On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

We prove that for a commutative ring R , every noetherian (artinian) R -module is quasi-injective if and only if every noetherian (artinian) R -module is quasi-projective if and only if the class of noetherian (artinian) R -modules is socle-fine if and only if the class of noetherian (artinian) R -modules is radical-fine if and only if every maximal ideal of R is idempotent.

On commutative rings whose prime ideals are direct sums of cyclics

M. Behboodi, A. Moradzadeh-Dehkordi (2012)

Archivum Mathematicum

In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring ( R , ) , the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R -modules; (2) = λ Λ R w λ where Λ is an index set and R / Ann ( w λ ) is a principal ideal ring for each λ Λ ; (3) Every prime ideal of R is a direct sum of at most...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

If A is a domain with the ascending chain condition on (integral) invertible ideals, then the group I ( A ) of its invertible ideals is generated by the set I m ( A ) of maximal invertible ideals. In this note we study some properties of I m ( A ) and we prove that, if I ( A ) is a free group on I m ( A ) , then A is a locally factorial Krull domain.

On endomorphisms of multiplication and comultiplication modules

H. Ansari-Toroghy, F. Farshadifar (2008)

Archivum Mathematicum

Let R be a ring with an identity (not necessarily commutative) and let M be a left R -module. This paper deals with multiplication and comultiplication left R -modules M having right End R ( M ) -module structures.

On finitely generated multiplication modules

R. Nekooei (2005)

Czechoslovak Mathematical Journal

We shall prove that if M is a finitely generated multiplication module and A n n ( M ) is a finitely generated ideal of R , then there exists a distributive lattice M ¯ such that S p e c ( M ) with Zariski topology is homeomorphic to S p e c ( M ¯ ) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R -modules M such that A n n ( M ) is a finitely generated ideal of R .

On n -submodules and G . n -submodules

Somayeh Karimzadeh, Javad Moghaderi (2023)

Czechoslovak Mathematical Journal

We investigate some properties of n -submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an n -submodule. Also, we show that if M is a finitely generated R -module and Ann R ( M ) is a prime ideal of R , then M has n -submodule. Moreover, we define the notion of G . n -submodule, which is a generalization of the notion of n -submodule. We find some characterizations of G . n -submodules and we examine the way the aforementioned notions are related to each...

On n-derivations and Relations between Elements rⁿ-r for Some n

Maciej Maciejewski, Andrzej Prószyński (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We find complete sets of generating relations between the elements [r] = rⁿ - r for n = 2 l and for n = 3. One of these relations is the n-derivation property [rs] = rⁿ[s] + s[r], r,s ∈ R.

On prime modules over pullback rings

Shahabaddin Ebrahimi Atani (2004)

Czechoslovak Mathematical Journal

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R -modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

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