The Kaplansky condition and rings of almost stable range $1$
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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
- Email: mroitman@math.haifa.ac.il
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3013-3018
- MSC (2010): Primary 13F99
- DOI: https://doi.org/10.1090/S0002-9939-2013-11567-4
- MathSciNet review: 3068954