A Bezout ring of stable range 2 which has square stable range 1. Issue 12 (2nd December 2019)

• Record Type: Journal Article
• Authors: Zabavsky, Bohdan; Romaniv, Oleh
• Is Part Of: Communications in algebra. Volume 47:Issue 12(2019)
• Page Start: 5392
• Page End: 5397
• Publication Date: 2019-12-02
• Subjects: Bezout ring -- Hermitian ring -- stable range 1 -- stable range 2 -- square stable range 1 -- neat range 1 -- elementary divisor ring -- Toeplitz matrix -- duo ring -- quasi-duo ring; 06F20 -- 13F99; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2019.1623239
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: In this article, we introduced the concept of a ring of stable range 2 which has square stable range 1. We proved that a Hermitian ring R which has (right) square stable range 1 is an elementary divisor ring if and only if R is a duo ring of neat range 1. And we found that a commutative Hermitian ring R is a Toeplitz ring if and only if R is a ring of (right) square range 1. We proved that if R be a commutative elementary divisor ring of (right) square stable range 1, then for any matrix A ∈ M 2 ( R ) one can find invertible Toeplitz matrices P and Q such that PAQ = ( e 1 0 0 e 2 ), where e 1 is a divisor of e 2 .