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* -biregular rings

Christopher J. Duckenfield (1972)

Commentationes Mathematicae Universitatis Carolinae

A minimal regular ring extension of C(X)

M. Henriksen, R. Raphael, R. G. Woods (2002)

Fundamenta Mathematicae

Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring C ( X , τ δ ) of continuous real-valued functions on the space ( X , τ δ ) , where τ δ is the smallest Tikhonov topology on X for which τ τ δ and C ( X , τ δ ) is von Neumann regular. The compact and metric spaces for which G ( X ) = C ( X , τ δ ) are characterized. Necessary, and different sufficient, conditions...

A note on orthodox additive inverse semirings

M. K. Sen, S. K. Maity (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We show in an additive inverse regular semiring ( S , + , · ) with E ( S ) as the set of all multiplicative idempotents and E + ( S ) as the set of all additive idempotents, the following conditions are equivalent: (i) For all e , f E ( S ) , e f E + ( S ) implies f e E + ( S ) . (ii) ( S , · ) is orthodox. (iii) ( S , · ) is a semilattice of groups. This result generalizes the corresponding result of regular ring.

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...

A Survey of Rings Generated by Units

Ashish K. Srivastava (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article presents a brief survey of the work done on rings generated by their units.

Adjoint regular rings.

Heatherly, Henry E., Tucci, Ralph P. (2002)

International Journal of Mathematics and Mathematical Sciences

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...

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