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.
Highlights
All rings are assumed to be commutative with unity 1
We generalize the concept of K-Hermite rings in the following sense: we call a ring R EM-Hermite, if for each a, b ∈ R, there exist a1, b1, d ∈ R such that a = a1d, b = b1d and the ideal (a1, b1) is regular
We find that this ring has some nice properties; it is preserved by the direct products and localizations, and unlike the case of K-Hermite rings, if R is EM-Hermite, so is R[x]
Summary
All rings are assumed to be commutative with unity 1. We generalize the concept of K-Hermite rings in the following sense: we call a ring R EM-Hermite, if for each a, b ∈ R, there exist a1, b1, d ∈ R such that a = a1d, b = b1d and the ideal (a1, b1) is regular. We find that this ring has some nice properties; it is preserved by the direct products and localizations, and unlike the case of K-Hermite rings, if R is EM-Hermite, so is R[x]. We characterize when an R-module with finite free resolution of length 1 is free
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