EM-HERMITE RINGS

A ring R is called 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 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 Mn,m(R) can be written as a product of a lower triangular matrix and a regular m × 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