Geodesic
Diagonal reductions of matrices over exchange ideals
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 9-18 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n(I)$, there exist left invertible $U_1,U_2\in M_n(R)$ and right invertible $V_1,V_2\in M_n(R)$ such that $U_1V_1AU_2V_2= \operatorname{diag}(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.
In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n(I)$, there exist left invertible $U_1,U_2\in M_n(R)$ and right invertible $V_1,V_2\in M_n(R)$ such that $U_1V_1AU_2V_2= \operatorname{diag}(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.
Classification : 15A21, 16D25, 16D70, 16E20, 16E50, 16U60, 16U99
Keywords: exchange ring; ideal; related comparability 
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a1/}
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Chen, Huanyin. Diagonal reductions of matrices over exchange ideals. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 9-18. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a1/

[1] P. Ara: Stability properties of exchange rings. Birkenmeier G. F. (ed.) et al., International symposium on ring theory. Boston, MA: Birkhäuser. Trends in Mathematics, 2001, pp. 23–42. | MR | Zbl

[2] 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. | MR

[3] 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. | DOI | MR

[4] P. Ara, G. K. Pedersen and F. Perera: An infinite analogue of rings with stable range one. J. Algebra 230 (2000), 608–655. | DOI | MR

[5] H. Chen: Elements in one-sided unit regular rings. Comm. Algebra 25 (1997), 2517–2529. | DOI | MR | Zbl

[6] H. Chen: Exchange rings, related comparability and power-substitution. Comm. Algebra 26 (1998), 3383–3668. | DOI | MR | Zbl

[7] H. Chen: Related comparability over exchange rings. Comm. Algebra 27 (1999), 4209–4216. | DOI | MR | Zbl

[8] H. Chen: Generalized stable regular rings. Comm. Algebra 31 (2003), 4899–4910. | DOI | MR | Zbl

[9] H. Chen and M. Chen: Generalized ideal-stable regular rings. Comm. Algebra 31 (2003), 4989–5001. | DOI | MR

[10] K. R. Goodearl: Von Neumann Regular Rings. Pitman, London, San Francisco, Melbourne, 1979, second ed., Krieger, Malabar, Fl., 1991. | MR | Zbl