Bass's first stable range condition

Abstract

Let R be an associative ring with unit. Bass’s lowest (the first) stable range condition [l] asserts the following: if a and b in R satisfy Ra + R& = R, then there exists C in R with a + tb left invertible (that is, R(a + tb) = R; Theorem 2.6 below says that the requirement that a + tb be a unit is not really stronger). More exactly, this is the ‘left’ version of the condition, and there is a symmetric ‘right’ version; but the two versions are in fact equivalent (see Theorem 2.1 below). For brevity, we call a ring satisfying the condition a B-ring. The expression ‘stable range (or rank) of R is l’, or ‘w(R) s I’, or ‘a ring of stable range 1’ is used for this in [19] and other places. In this note known results are collected and a little information is added on Brings. We do not introduce here Bass’s higher stable range conditions (see [I, 3, 9, 11, 13, 14, 16-20, 221) which follow from the first one by [19, Theorem 11, or nonassociative B-rings (see [4, 51, where ‘ring of stable range 2’ means a ring satisfying the first stable range condition). A nice geometric characterization of B-rings is given in [21]. The note was inspired by (and called after) an old unpublished paper by I. Kaplansky which contains Theorems 2.4, 2.6, 2.8, and 5.3, so I dedicate it to him. I thank him for permition to publish his results and pointing out the paper [6] (which contains Theorems 2.2, 2.7, and the equivalence of 5.3(a) with 5.3(d); a ring has ‘substitution property’ in the sense of [6] if and only if it is a B-ring), R. Herman for discussions of the stable range of P-algebras, and a referee for suggestions.