EM-HERMITE RINGS

Year 2020, Volume: 27 Issue: 27, 88 - 101, 07.01.2020

Emad Abuosba Manal Ghanem

https://doi.org/10.24330/ieja.662967

Cited By: 1

Abstract

A ring $R$ is called EM-Hermite if for each $a,b\in R$, there exist $%
a_{1},b_{1},d\in R$ such that $a=a_{1}d,b=b_{1}d$ and the ideal $%
(a_{1},b_{1})$ is regular. We give several characterizations of
EM-Hermite rings analogue to those for K-Hermite rings, for
example, $R$ is an EM-Hermite ring if and only if any matrix in
$M_{n,m}(R)$ can be written as a product of a lower triangular
matrix and a regular $m\times m$ matrix. We relate EM-Hermite
rings to Armendariz rings, rings with a.c. condition, rings with
property A, EM-rings, generalized morphic rings, and PP-rings. We
show that for an EM-Hermite ring, the polynomial ring and
localizations are also EM-Hermite rings, and show that any regular
row can be extended to regular matrix. We relate EM-Hermite rings
to weakly semi-Steinitz rings, and characterize the case at which
every finitely generated $R$-module with
finite free resolution of length 1 is free.

Keywords

Hermite ring, K-Hermite ring, weakly semi-Steinitz ring, generalized morphic ring, regular matrix

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