EM-HERMITE RINGS

Authors : Emad ABUOSBA, Manal GHANEM

Pages : 88-101

Doi:10.24330/ieja.662967

View : 79 | Download : 11

Publication Date : 2020-01-07

Article Type : Research Paper

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 $% insert ignore into journalissuearticles values(a_{1},b_{1});$ 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 $M_{n,m}insert ignore into journalissuearticles values(R);$ 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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