Commutative Bezout domains of stable range 1.5

2018/06/12 by Bovdi, Victor A., Shchedryk, Volodymyr P. · 2 citations
#FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1806.04747

Abstract

A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R.

Cited by

• Elementary divisor rings with Dubrovin-Komarnytsky property
• Rings of the right (left) almost stable range 1

Related

• Elementary matrix reduction over locally stable rings
• Rings of the right (left) almost stable range 1
• Conditions for stable range of an elementary divisor rings
• A Bezout ring of stable range 2 which has square stable range 1
• A commutative Bezout domain in which every maximal ideal is principal is\n an elementary divisor ring