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Closure rings

Barry J. Gardner, Tim Stokes (1999)

Commentationes Mathematicae Universitatis Carolinae

We consider rings equipped with a closure operation defined in terms of a collection of commuting idempotents, generalising the idea of a topological closure operation defined on a ring of sets. We establish the basic properties of such rings, consider examples and construction methods, and then concentrate on rings which have a closure operation defined in terms of their lattice of central idempotents.

Commutativity of rings with constraints involving a subset

Moharram A. Khan (2003)

Czechoslovak Mathematical Journal

Suppose that R is an associative ring with identity 1 , J ( R ) the Jacobson radical of R , and N ( R ) the set of nilpotent elements of R . Let m 1 be a fixed positive integer and R an m -torsion-free ring with identity 1 . The main result of the present paper asserts that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y R J ( R ) and (ii) [ ( x y ) m + y m x m , x ] = 0 = [ ( y x ) m + x m y m , x ] , for all x , y R J ( R ) . This result is also valid if (i) and (ii) are replaced by (i) ' ...

Commutativity of rings with polynomial constraints

Moharram A. Khan (2002)

Czechoslovak Mathematical Journal

Let p , q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s -unital ring satisfying [ f ( x p y q ) - x r y , x ] = 0 ( [ f ( x p y q ) - y x r , x ] = 0 , respectively) where f ( λ ) λ 2 [ λ ] , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

Diagonal reductions of matrices over exchange ideals

Huanyin Chen (2006)

Czechoslovak Mathematical Journal

In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R . If I satisfies related comparability, then for any regular matrix A M n ( I ) , there exist left invertible U 1 , U 2 M n ( R ) and right invertible V 1 , V 2 M n ( R ) such that U 1 V 1 A U 2 V 2 = diag ( e 1 , , e n ) for idempotents e 1 , , e n I .

Exchange rings in which all regular elements are one-sided unit-regular

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

Exchange rings with stable range one

Huanyin Chen (2007)

Czechoslovak Mathematical Journal

We characterize exchange rings having stable range one. An exchange ring R has stable range one if and only if for any regular a R , there exist an e E ( R ) and a u U ( R ) such that a = e + u and a R e R = 0 if and only if for any regular a R , there exist e r . a n n ( a + ) and u U ( R ) such that a = e + u if and only if for any a , b R , R / a R R / b R a R b R .

Extensions of G M -rings

Huanyin Chen, Miaosen Chen (2005)

Czechoslovak Mathematical Journal

It is shown that a ring R is a G M -ring if and only if there exists a complete orthogonal set { e 1 , , e n } of idempotents such that all e i R e i are G M -rings. We also investigate G M -rings for Morita contexts, module extensions and power series rings.

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