## Taiwanese Journal of Mathematics

### WEAKLY STABLE IDEALS OF EXCHANGE RINGS

Huanyin Chen

#### Abstract

Let I be an ideal of an exchange ring R. We say that I is weakly stable provided that every regular element in 1+I is one-sided unit-regular. If I is weakly stable, then so is Mn(I) as an ideal of Mn(R). Also every square regular matrix over such an ideal admits a diagonal reduction by right or left invertible matrices. These extend the corresponding results of [1],[6-8],[10] and [12-13].

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 1 (2008), 25-38.

Dates
First available in Project Euclid: 21 July 2017

https://projecteuclid.org/euclid.twjm/1500602486

Digital Object Identifier
doi:10.11650/twjm/1500602486

Mathematical Reviews number (MathSciNet)
MR2387102

Zentralblatt MATH identifier
1152.16008

Keywords
exchange ring ideal comparability

#### Citation

Chen, Huanyin. WEAKLY STABLE IDEALS OF EXCHANGE RINGS. Taiwanese J. Math. 12 (2008), no. 1, 25--38. doi:10.11650/twjm/1500602486. https://projecteuclid.org/euclid.twjm/1500602486

#### References

• [1.] P. Ara, K. R. Goodearl, K. C. O'Meara and E. Pardo, Diagonalization of matrices over regular rings, Linear Algebra Appl., 265 (1997), 147-163.
• [2.] P. Ara, K. R. Goodearl, K. C. O'Meara and E.Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math., 105 (1998), 105-137.
• [3.] P. Ara, G. K. Pedersen, F. Perera, An Infinite analogue of rings with stable range one, J. Algebra, 230 (2000), 608-655.
• [4.] H. Chen, Elements in one-sided unit regular rings, Comm. Algebra, 25 (1997), 2517-2529.
• [5.] H. Chen, Comparability of modules over regular rings, Comm. Algebra,25 (1997), 3531-3543.
• [6.] H. Chen, Related comparability over exchange rings, Comm. Algebra, 27 (1999), 4209-4216.
• [7.] H. Chen, One-sided unit-regular ideals of regular rings, Taiwanese J. Math., 8 (2004), 761-770.
• [8.] Henriksen, M. On a class of regular rings that are elementary divisor rings. Arch. Math., 1973, 24, 133-141.
• [9.] T. Y. Lam, A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl., 3 (2004), 301-343.
• [10.] Q. Li, Comparability, $K_0$ and Its State Space, Ph.D. Thesis, Nanjing University, China, 2002.
• [11.] A. A. Tuganbaev, Rings Close to Regular, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002.
• [12.] J. Wei, Exchange rings with weakly stable range one, Vietnam J. Math., 32(2004), 441-449.
• [13.] J. Wei, Unit-regularity and stable range conditions, Comm. Algebra, 33 (2005), 1937-1946.