Elementary matrix reduction over certain rings

摘要

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR + bR = R double right arrow there exists x is an element of R such that R/(a + bx) R has stable range 1. We study locally stable rings and prove that every locally stable Bezout ring is an elementary divisor ring. Many known results on domains are thereby generalized.