Elementary matrix reduction over J-stable rings

Abstract

ABSTRACTA commutative ring R is J-stable provided that R∕aR has stable range 1 for all a∉J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3, Theorem 8], [4, Theorem 4.1], [7, Theorem 3.7], [8, Theorem], [9, Theorem 2.1], [14, Theorem 1] and [18, Theorem 7].