ABSTRACT
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⇒∃x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.
Acknowledgement
The authors are grateful to the referee for his/her helpful suggestions which make the new version more clearer. For instance, the proof of Theorem 2.2. H. Chen is thankful for the support by the Natural Science Foundation of Zhejiang Province, China (LY17A010018).