Stable range one for rings with many units

Abstract

AbstractThe main purpose of this paper is to prove the stable range 1 condition for a number of classes of rings and algebras. Using a modification of a computation of D.V. Tyukavkin, stable range 1 (and a bit more) is obtained from the following simple condition on a ring R: given any x, y ϵ R, there is a unit u ϵ R such that x − u and y − u-1 are both units. Verification of the latter condition then yields stable range 1 in a number of cases, e.g.: (1) any algebra over an uncountable field, in which all non-zero-divisors are units and there are no uncountable direct sums of nonzero one-sided ideals; (2) any algebra over an uncountable field, with only countably many primitive factor rings, all of which are artinian; (3) the endomorphism ring of any noetherian module over an algebra as in (2); (4) any algebraic algebra over an infinite field; (5) any integral algebra over a commutative ring which modulo its Jacobson radical is algebraic over an infinite field; (6) any von Nuemann regular algebra over an uncountable field, which has a rank function. Using other techniques, it is proved that finite Rickart C∗-algebras, strongly π-regular von Neumann regular rings, and strongly π-regular rings in which every element is a sum of a unit plus a central unit, all have stable range 1. Finally, for an arbitrary commutative ring some overrings with specified stable range properties are constructed, in particular a more or less canonical overring having stable range 1