Abstract
References
- E. Abuosba and M. Ghanem, Annihilating content in polynomial and power series rings, J. Korean Math. Soc., 56(5) (2019), 1403-1418.
- 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.
- W. C. Brown, Matrices over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 169, Marcel Dekker, Inc., New York, 1993.
- S. Endo, Note on p.p. rings, A supplement to Hattori's paper, Nagoya Math. J., 17 (1960), 167-170.
- 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.
- L. Gillman and M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc., 82 (1956), 362-365.
- M. Henriksen, Some remarks on elementary divisor rings II, Michigan Math. J., 3(2) (1955), 159-163.
- I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
- I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago, 1974.
- T. Y. Lam, Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
- 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.
- B. Nashier and W. Nichols, On Steinitz properties, Arch. Math. (Basel), 57(3) (1991), 247-253.
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
- E. Abuosba and M. Ghanem, Annihilating content in polynomial and power series rings, J. Korean Math. Soc., 56(5) (2019), 1403-1418.
- 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.
- W. C. Brown, Matrices over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 169, Marcel Dekker, Inc., New York, 1993.
- S. Endo, Note on p.p. rings, A supplement to Hattori's paper, Nagoya Math. J., 17 (1960), 167-170.
- 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.
- L. Gillman and M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc., 82 (1956), 362-365.
- M. Henriksen, Some remarks on elementary divisor rings II, Michigan Math. J., 3(2) (1955), 159-163.
- I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
- I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago, 1974.
- T. Y. Lam, Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
- 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.
- B. Nashier and W. Nichols, On Steinitz properties, Arch. Math. (Basel), 57(3) (1991), 247-253.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Articles |
| Authors | |
| Publication Date | January 7, 2020 |
| Published in Issue | Year 2020 |