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The Kaplansky condition and rings of almost stable range $ 1$

Author: Moshe Roitman
Journal: Proc. Amer. Math. Soc. 141 (2013), 3013-3018
MSC (2010): Primary 13F99
Published electronically: May 22, 2013
MathSciNet review: 3068954
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Abstract: We present some variants of the Kaplansky condition for a K-Hermite ring $ R$ to be an elementary divisor ring. For example, a commutative K-Hermite ring $ R$ is an EDR iff for any elements $ x,y,z\in R$ such that $ (x,y)=R$ there exists an element $ \lambda \in R$ such that $ x+\lambda y=uv$, where $ (u,z)=(v,1-z)=R$.

We present an example of a Bézout domain that is an elementary divisor ring but does not have almost stable range $ 1$, thus answering a question of Warren Wm. McGovern.

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Additional Information

Moshe Roitman
Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Keywords: Almost stable range $1$, elementary divisor ring, Kaplansky condition, K-Hermite, stable range
Received by editor(s): July 15, 2011
Received by editor(s) in revised form: November 24, 2011
Published electronically: May 22, 2013
Additional Notes: Part of this work was done while the author was visiting New Mexico State University. The author thanks Bruce Olberding from this university for useful discussions and suggestions concerning this topic
Communicated by: Irena Peeva
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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