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A generalization of reflexive rings

Mete Burak Çalcı, Huanyin Chen, Sait Halıcıoğlu (2024)

Mathematica Bohemica

We introduce a class of rings which is a generalization of reflexive rings and J -reversible rings. Let R be a ring with identity and J ( R ) denote the Jacobson radical of R . A ring R is called J -reflexive if for any a , b ∈ R , a R b = 0 implies b R a ⊆ J ( R ) . We give some characterizations of a J -reflexive ring. We prove that some results of reflexive rings can be extended to J -reflexive rings for this general setting. We conclude some relations between J -reflexive rings and some related rings. We investigate some extensions of...

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

In this paper, we introduce a subclass of strongly clean rings. Let R be a ring with identity, J be the Jacobson radical of R , and let J # denote the set of all elements of R which are nilpotent in R / J . An element a ∈ R is called very J # -clean provided that there exists an idempotent e ∈ R such that a e = e a and a - e or a + e is an element of J # . A ring R is said to be very J # -clean in case every element in R is very J # -clean. We prove that every very J # -clean ring is strongly π -rad clean and has stable range one. It is shown...

Almost Abelian rings

Junchao Wei (2013)

Communications in Mathematics

A ring R is defined to be left almost Abelian if a e = 0 implies a R e = 0 for a ∈ N ( R ) and e ∈ E ( R ) , where E ( R ) and N ( R ) stand respectively for the set of idempotents and the set of nilpotents of R . Some characterizations and properties of such rings are included. It follows that if R is a left almost Abelian ring, then R is π -regular if and only if N ( R ) is an ideal of R and R / N ( R ) is regular. Moreover it is proved that (1) R is an Abelian ring if and only if R is a left almost Abelian left idempotent reflexive ring. (2) R is strongly...

Anneaux semi-artiniens

Constantin Năstăsescu, Nicolae Popescu (1968)

Bulletin de la Société Mathématique de France

Closed extensions of R-modules in the case of a semi-artinian ring R

Frans Loonstra (1985)

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

Si considerano le estensioni chiuse B di un R -modulo A mediante un R -modulo C nel caso in cui R sia un anello semi-artiniano, cioè un anello R con la proprietà che per ogni quoziente ( R / I ) ≠ 0 sia soc ( R / I ) ≠ 0 . Tali estensioni sono caratterizzate dal fatto che A deve essere un sottomodulo semi-puro di B .

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

Farid Kourki, Rachid Tribak (2023)

Czechoslovak Mathematical Journal

We provide some characterizations of rings R for which every (finitely generated) module belonging to a class 𝒞 of R -modules is a direct sum of cyclic submodules. We focus on the cases, where the class 𝒞 is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.

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