2×2 INVERTIBLE MATRICES OVER WEAKLY STABLE RINGS

Title & Authors
2×2 INVERTIBLE MATRICES OVER WEAKLY STABLE RINGS
Chen, Huanyin;

Abstract
A ring R is a weakly stable ring provided that aR + bR = R implies that there exists $\small{y\;{\in}\;R}$ such that $\small{a\;+\;by\;{\in}\;R}$ is right or left invertible. In this article, we characterize weakly stable rings by virtue of $\small{2{\times}2}$ invertible matrices over them. It is shown that a ring R is a weakly stable ring if and only if for any $\small{A\;{\in}GL_2(R)}$, there exist two invertible lower triangular L and K and an invertible upper triangular U such that A = LUK, where two of L, U and K have diagonal entries 1. Related results are also given. These extend the work of Nagarajan et al.ਆጊ尀Ѐ㘶ㄻԀ䭃䑎䴀
Keywords
weakly stable ring;invertible matrix;factorization;
Language
English
Cited by
1.
ON QUASI-STABLE EXCHANGE IDEALS,;

대한수학회지, 2010. vol.47. 1, pp.1-15
2.
PIERCE STALKS OF EXCHANGE RINGS,;

대한수학회지, 2010. vol.47. 4, pp.819-830
1.
PIERCE STALKS OF EXCHANGE RINGS, Journal of the Korean Mathematical Society, 2010, 47, 4, 819
References
1.
P. Ara, The exchange property for purely infinite simple rings, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2543–2547.

2.
H. Chen, Comparability of modules over regular rings, Comm. Algebra 25 (1997), no. 11, 3531–3543.

3.
H. Chen, Exchange rings in which all regular elements are one-sided unit-regular, Czechoslovak Math. J. 58 (2008), no. 4, 899–910.

4.
H. Chen and M. Chen, On products of three triangular matrices over associative rings, Linear Algebra Appl. 387 (2004), 297–311.

5.
H. Chen and M. Chen, On strongly π-regular ideals, J. Pure Appl. Algebra 195 (2005), no. 1, 21–32.

6.
G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc. 216 (1976), 81–90.

7.
K. R. Goodearl, von Neumann Regular Rings, Second edition. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.

8.
T. Y. Lam, A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl. 3 (2004), no. 3, 301–343.

9.
Q. Li and W. Tong, Weak cancellation of modules and weakly stable range conditions for exchange rings, Acta Math. Sinica (Chin. Ser.) 45 (2002), no. 6, 1121–1126.

10.
D. Lu, Q. Li, and W. Tong, Comparability, stability, and completions of ideals, Comm. Algebra 32 (2004), no. 7, 2617–2634.

11.
K. R. Nagarajan, M. D. Devasahayam, and T. Soundararajan, Products of three triangular matrices, Linear Algebra Appl. 292 (1999), no. 1-3, 61–71.

12.
K. R. Nagarajan, M. D. Devasahayam, and T. Soundararajan, Products of three triangular matrices over commutative rings, Linear Algebra Appl. 348 (2002), 1–6.

13.
A. A. Tuganbaev, Rings Close to Regular, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002.

14.
L. N. Vaserstein and E. Wheland, Commutators and companion matrices over rings of stable rank 1, Linear Algebra Appl. 142 (1990), 263–277.

15.
J. Wei, Exchange rings with weakly stable range one, Vietnam J. Math. 32 (2004), no. 4, 441–449.

16.
J. Wei, Unit-regularity and stable range conditions, Comm. Algebra 33 (2005), no. 6, 1937–1946.