A note on regular rings with stable range one.

Chen, H.V.; Chin, A.Y.M.

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Exchange rings having stable range one.

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Exchange rings with stable range one

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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 $a \in R$ , there exist an $e \in E (R)$ and a $u \in U (R)$ such that $a = e + u$ and $a R \cap e R = 0$ if and only if for any regular $a \in R$ , there exist $e \in r . a n n (a^{+})$ and $u \in U (R)$ such that $a = e + u$ if and only if for any $a, b \in R$ , $R / a R ≅ R / b R ⟹ a R ≅ b R$ .

The almost isomorphism relation for simple regular rings.

Pere Ara, Kenneth R. Goodearl (1992)

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A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K. C. O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring.

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Exchange rings satisfying the related comparability.

Huanyin Chen, Fu-An Li (2002)

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In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.