The Kaplansky condition and rings of almost stable range
Author:
Moshe Roitman
Journal:
Proc. Amer. Math. Soc. 141 (2013), 30133018
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 KHermite ring to be an elementary divisor ring. For example, a commutative KHermite 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 𝐾theory, W. A. Benjamin, Inc., New
YorkAmsterdam, 1968. MR 0249491
(40 #2736)
 [2]
Douglas
Costa, Joe
L. Mott, and Muhammad
Zafrullah, The construction
𝐷+𝑋𝐷_{𝑠}[𝑋], J. Algebra
53 (1978), no. 2, 423–439. MR 0506224
(58 #22046)
 [3]
László
Fuchs and Luigi
Salce, Modules over nonNoetherian 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), 10.1090/S00029947195600789804
 [5]
Leonard
Gillman and Melvin
Henriksen, Some remarks about elementary divisor
rings, Trans. Amer. Math. Soc. 82 (1956), 362–365. MR 0078979
(18,9c), 10.1090/S00029947195600789798
 [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), 10.1016/j.jalgebra.2007.01.007
 [8]
Irving
Kaplansky, Elementary divisors and
modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. MR 0031470
(11,155b), 10.1090/S00029947194900314703
 [9]
T.
Y. Lam, Serre’s problem on projective modules, Springer
Monographs in Mathematics, SpringerVerlag, Berlin, 2006. MR 2235330
(2007b:13014)
 [10]
Warren
Wm. McGovern, Bézout rings with almost stable range 1,
J. Pure Appl. Algebra 212 (2008), no. 2,
340–348. MR 2357336
(2008h:13033), 10.1016/j.jpaa.2007.05.026
 [11]
Pere
Menal and Jaume
Moncasi, On regular rings with stable range 2, J. Pure Appl.
Algebra 24 (1982), no. 1, 25–40. MR 647578
(83g:16025), 10.1016/00224049(82)900561
 [12]
B.
V. Zabavs′ki&ibreve;, Reduction of matrices over Bezout rings
of stable rank at most 2, Ukraïn. Mat. Zh. 55
(2003), no. 4, 550–554 (Ukrainian, with English and Ukrainian
summaries); English transl., Ukrainian Math. J. 55
(2003), no. 4, 665–670. MR
2072558, 10.1023/B:UKMA.0000010166.70532.41
 [1]
 Hyman Bass, Algebraic theory, W. A. Benjamin, Inc., New YorkAmsterdam, 1968. MR 0249491 (40:2736)
 [2]
 Douglas Costa, Joe L. Mott, and Muhammad Zafrullah, The construction , J. Algebra 53 (1978), no. 2, 423439. MR 0506224 (58:22046)
 [3]
 László Fuchs and Luigi Salce, Modules over nonNoetherian 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), 366391. MR 0078980 (18:9d)
 [5]
 Leonard Gillman and Melvin Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362365. MR 0078979 (18:9c)
 [6]
 Melvin Henriksen, Some remarks on elementary divisor rings. II, Michigan Math. J. 3 (19551956), 159163. MR 0092772 (19:1155i)
 [7]
 Evan Houston and John Taylor, Arithmetic properties in pullbacks, J. Algebra 310 (2007), no. 1, 235260. MR 2307792 (2008b:13028)
 [8]
 Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464491. MR 0031470 (11:155b)
 [9]
 T. Y. Lam, Serre's problem on projective modules, Springer Monographs in Mathematics, SpringerVerlag, Berlin, 2006. MR 2235330 (2007b:13014)
 [10]
 Warren Wm. McGovern, Bézout rings with almost stable range , J. Pure Appl. Algebra 212 (2008), no. 2, 340348. MR 2357336 (2008h:13033)
 [11]
 Pere Menal and Jaume Moncasi, On regular rings with stable range , J. Pure Appl. Algebra 24 (1982), no. 1, 2540. 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, 550554. MR 2072558
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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
DOI:
http://dx.doi.org/10.1090/S000299392013115674
PII:
S 00029939(2013)115674
Keywords:
Almost stable range $1$,
elementary divisor ring,
Kaplansky condition,
KHermite,
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.
