EM-HERMITE RINGS

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

References

1. E. Abuosba and M. Ghanem, Annihilating content in polynomial and power series rings, J. Korean Math. Soc., 56(5) (2019), 1403-1418.
2. A. Bouanane and F. Kourki, On weakly semi-Steinitz rings, Commutative Ring Theory, Lecture Notes in Pure and Appl. Math., Dekker, New York, 185 (1997), 131-139.
3. W. C. Brown, Matrices over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 169, Marcel Dekker, Inc., New York, 1993.
4. S. Endo, Note on p.p. rings, A supplement to Hattori's paper, Nagoya Math. J., 17 (1960), 167-170.
5. L. Gillman and M. Henriksen, Rings of continuous functions in which every nitely generated ideal is principal, Trans. Amer. Math. Soc., 82 (1956), 366- 391.
6. L. Gillman and M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc., 82 (1956), 362-365.
7. M. Henriksen, Some remarks on elementary divisor rings II, Michigan Math. J., 3(2) (1955), 159-163.
8. I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.

9. I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago, 1974.
10. T. Y. Lam, Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
11. M. D. Larsen, W. J. Lewis and T. S. Shores, Elementary divisor rings and nitely presented modules, Trans. Amer. Math. Soc., 187 (1974), 231-248.
12. B. Nashier and W. Nichols, On Steinitz properties, Arch. Math. (Basel), 57(3) (1991), 247-253.