A class of unitary modules with stable range 2

Abstract

The notion and some properties of B-rings, in a natural way, are extended to B-stable and BJ-stable modules. Let J(R) be the Jacobson radical of a commutative ring R with identity 1 ≠ 0. A sequence (a1, …, an+1), n ≥ 1, of elements in R is said to be unimodular if 1 (Formula present). A ring R is said to be a B-ring if for any unimodular sequence (Formula present), there exists an element b in R such that (Formula present). Our aim in this paper is to extend this notion in the context of modules. It is shown that a cyclic module over a B-ring (in particular, semilocal ring, Noetherian ring in which every prime ideal is maximal, Dedekind domain) is BJ-stable. Also, a torsion-free cyclic R-module is B-stable if and only if R is a B-stable ring. Any module of rank greater than or equal to 3 is never BJ-stable and consequently not B-stable. We show that a finitely generated R-module A is BJ-stable if Z(B) is finite for every submodule B of A with B ⊈ J(A), where Z(B) denotes the set of all maximal submodules of A containing B.