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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 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.

On prime submodules and primary decomposition

Yücel Tiraş, Harmanci, Abdullah (2000)

Czechoslovak Mathematical Journal

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R .

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