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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 .

FC-modules with an application to cotorsion pairs

Yonghua Guo (2009)

Commentationes Mathematicae Universitatis Carolinae

Let R be a ring. A left R -module M is called an FC-module if M + = Hom ( M , / ) is a flat right R -module. In this paper, some homological properties of FC-modules are given. Let n be a nonnegative integer and ℱ𝒞 n the class of all left R -modules M such that the flat dimension of M + is less than or equal to n . It is shown that ( ( ℱ𝒞 n ) , ℱ𝒞 n ) is a complete cotorsion pair and if R is a ring such that fd ( ( R R ) + ) n and ℱ𝒞 n is closed under direct sums, then ( ℱ𝒞 n , ℱ𝒞 n ) is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....

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