On rigid rings
type: Item
instance of: scholarly article
title: On rigid rings
zbMATH Open document ID: 0810.13021
DOI: 10.1016/0022-4049(94)90075-2
author: Pramod K. Sharma
published in: Journal of Pure and Applied Algebra
publication date: 1994-11-03T00:00:00Z
review text: A ring \(R\) is said to be rigid if whenever for a submodule \(K\) of a free \(R\)-module \(F\) of rank \(n\), \(F/K\) is generated by \(m<n\) elements, \(K\) contains a unimodular element of \(R\). This concept originated in the study of pole-assignment of linear systems over rings, but the author here shows that it is also interesting algebraically. Several classes of rings are shown to be rigid, for example BCU-rings, homomorphic images of rigid rings, power series rings of rigid rings, products of rigid rings, semi-local rings, PIDs etc. Sufficient conditions for \(R[X]\) to be rigid are also given. The paper is concluded by several examples of non-rigid rings.
Mathematics Subject Classification ID: 13F99
Mathematics Subject Classification ID: 13F50
Mathematics Subject Classification ID: 13F45
zbMATH DE Number: 681820
zbMATH Keywords: pole-assignment of linear systems
zbMATH Keywords: rigid rings
zbMATH Keywords: power series rings
reviewed by: Frieda Theron
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90075-2
OpenAlex ID: W2039366557
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Some cancellation theorems about projective modules over polynomial rings
cites work: Stability theorems for overrings of polynomial rings
cites work: Pole assignability in polynomial rings, power series rings, and Prüfer domains
cites work: Polynomial rings which are BCS rings
cites work: Remarks on the pole-shifting problem over rings
cites work: Generating modules efficiently: Theorems from algebraic K-theory
cites work: Module equivalences: Local to global. When primitive polynomials represent units
cites work: Comaximizable Primes
cites work: Q3739243
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Comments on pole assignability over rings
cites work: Q4092100
cites work: Stability theorems for overrings of polynomial rings. II
cites work: Some results on pole-placement and reachability
cites work: Q4124941
cites work: BCS rings
description: scientific article; zbMATH DE number 681820