ON m-FOLD STABLE IDEALS

Abstract

In this paper, we introduce m-fold stable ideals and show that m-fold stable range for ideals is invariant under matrix extension. Also we prove that every m-fold stable ideal is right and left symmetric.Let R be an associative ring with identity. We say that R is an m-fold stable ring in case when a 1 R + b 1 R = R, . . . , a m R + b m R = R, there exists a y ∈ R such that a 1 + b 1 y, . . . , a m + b m y ∈ U (R). The m-fold stable range condition is very useful in algebra K-theory (see [3]). The main purpose of this paper is to extend the m-fold stable range condition to ideals. Let I be an ideal of a ring R. We say that I is m-fold stable provided that a m ∈ 1 + I and b 1 , . . . , b m ∈ I implies that a 1 + b 1 y, . . . , a m + b m y ∈ U (R) for a y ∈ R. Clearly, a ring R is an m-fold stable ring if and only if R is an mfold stable ideal. Let R = Z/2Z ⊕ Z/3Z and I = 0 ⊕ Z/3Z. Then R is not 2-fold stable, while I is 2-fold stable as an ideal of R. Thus we see that m-fold stable ideals are nontrivial generalizations of m-fold stable rings.Throughout, rings are associative with identities and modules are right modules. GL n (R) denotes the general linear group of R, U (R) denotes the set of units of R, I n denotes the n by n matrix diag(1, . . . , 1). Set B ij (x) = I 2 + xe ij (i = j, 1 < i, j < 2), [α, β] = αe 11 + βe 22 , where e ij (1 ≤ i, j ≤ 2) are all matrix units, x ∈ R and α, β ∈ U (R). Lemma 1. Let I be an ideal of a ring R. Then the following are equivalent: (1) I is m-fold stable. 239 J. Algebra Appl. 2004.03:239-246. Downloaded from www.worldscientific.com by AUCKLAND UNIVERSITY OF TECHNOLOGY on 03/09/15. For personal use only.

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