Displaying similar documents to “Diagonal reductions of matrices over exchange ideals”

On rings close to regular and p -injectivity

Roger Yue Chi Ming (2006)

Commentationes Mathematicae Universitatis Carolinae

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The following results are proved for a ring A : (1) If A is a fully right idempotent ring having a classical left quotient ring Q which is right quasi-duo, then Q is a strongly regular ring; (2) A has a classical left quotient ring Q which is a finite direct sum of division rings iff A is a left TC -ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let A have the following properties: (a) each maximal left ideal of A is either a two-sided...

Strong separativity over exchange rings

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

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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.

The maximal regular ideal of some commutative rings

Emad Abu Osba, Melvin Henriksen, Osama Alkam, Frank A. Smith (2006)

Commentationes Mathematicae Universitatis Carolinae

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In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring R has an ideal 𝔐 ( R ) consisting of elements a for which there is an x such that a x a = a , and maximal with respect to this property. Considering only the case when R is commutative and has an identity element, it is often not easy to determine when 𝔐 ( R ) is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of a or 1 - a has a von...

Exchange rings with stable range one

Huanyin Chen (2007)

Czechoslovak Mathematical Journal

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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 .