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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 .

Strong separativity over exchange rings

Huanyin Chen — 2008

Czechoslovak Mathematical Journal

An exchange ring R is strongly separative provided that for all finitely generated projective right R -modules A and B , A A A B A B . We prove that an exchange ring R is strongly separative if and only if for any corner S of R , a S + b S = S implies that there exist u , v S such that a u = b v and S u + S v = S if and only if for any corner S of R , a S + b S = S implies that there exists a right invertible matrix a b * M 2 ( S ) . The dual assertions are also proved.

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 .

Clean matrices over commutative rings

Huanyin Chen — 2009

Czechoslovak Mathematical Journal

A matrix A M n ( R ) is e -clean provided there exists an idempotent E M n ( R ) such that A - E GL n ( R ) and det E = e . We get a general criterion of e -cleanness for the matrix [ [ a 1 , a 2 , , a n + 1 ] ] . Under the n -stable range condition, it is shown that [ [ a 1 , a 2 , , a n + 1 ] ] is 0 -clean iff ( a 1 , a 2 , , a n + 1 ) = 1 . As an application, we prove that the 0 -cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n 3 . The analogous for ( s , 2 ) property is also obtained.

Extensions of G M -rings

Huanyin ChenMiaosen 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.

Certain decompositions of matrices over Abelian rings

Nahid AshrafiMarjan SheibaniHuanyin Chen — 2017

Czechoslovak Mathematical Journal

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n . We prove that M n ( R ) is nil clean if and only if R / J ( R ) is Boolean and M n ( J ( R ) ) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R / J ( R ) is 3 , B or 3 B where B is a Boolean ring, and that M n ( R ) is weakly nil clean if and only if M n ( R ) is nil clean for all n 2 .

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