The Kaplansky condition and rings of almost stable range

Author:
Moshe Roitman

Journal:
Proc. Amer. Math. Soc. **141** (2013), 3013-3018

MSC (2010):
Primary 13F99

DOI:
https://doi.org/10.1090/S0002-9939-2013-11567-4

Published electronically:
May 22, 2013

MathSciNet review:
3068954

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present some variants of the Kaplansky condition for a K-Hermite ring to be an elementary divisor ring. For example, a commutative K-Hermite ring is an EDR iff for any elements such that there exists an element such that , where .

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

**[1]**Hyman Bass,*Algebraic -theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0249491 (40:2736)****[2]**Douglas Costa, Joe L. Mott, and Muhammad Zafrullah,*The construction*, J. Algebra**53**(1978), no. 2, 423-439. MR**0506224 (58:22046)****[3]**László Fuchs and Luigi Salce,*Modules over non-Noetherian domains*, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. MR**1794715 (2001i:13002)****[4]**Leonard Gillman and Melvin Henriksen,*Rings of continuous functions in which every finitely generated ideal is principal*, Trans. Amer. Math. Soc.**82**(1956), 366-391. MR**0078980 (18:9d)****[5]**Leonard Gillman and Melvin Henriksen,*Some remarks about elementary divisor rings*, Trans. Amer. Math. Soc.**82**(1956), 362-365. MR**0078979 (18:9c)****[6]**Melvin Henriksen,*Some remarks on elementary divisor rings. II*, Michigan Math. J.**3**(1955-1956), 159-163. MR**0092772 (19:1155i)****[7]**Evan Houston and John Taylor,*Arithmetic properties in pullbacks*, J. Algebra**310**(2007), no. 1, 235-260. MR**2307792 (2008b:13028)****[8]**Irving Kaplansky,*Elementary divisors and modules*, Trans. Amer. Math. Soc.**66**(1949), 464-491. MR**0031470 (11:155b)****[9]**T. Y. Lam,*Serre's problem on projective modules*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. MR**2235330 (2007b:13014)****[10]**Warren Wm. McGovern,*Bézout rings with almost stable range*, J. Pure Appl. Algebra**212**(2008), no. 2, 340-348. MR**2357336 (2008h:13033)****[11]**Pere Menal and Jaume Moncasi,*On regular rings with stable range*, J. Pure Appl. Algebra**24**(1982), no. 1, 25-40. MR**647578 (83g:16025)****[12]**B. V. Zabavskiĭ,*Reduction of matrices over Bezout rings of stable rank at most*, Ukraïn. Mat. Zh.**55**(2003), no. 4, 550-554. MR**2072558**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
13F99

Retrieve articles in all journals with MSC (2010): 13F99

Additional Information

**Moshe Roitman**

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

Email:
mroitman@math.haifa.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-2013-11567-4

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.