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Year 2020, , 88 - 101, 07.01.2020
https://doi.org/10.24330/ieja.662967

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.

EM-HERMITE RINGS

Year 2020, , 88 - 101, 07.01.2020
https://doi.org/10.24330/ieja.662967

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.

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.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Emad Abuosba This is me

Manal Ghanem

Publication Date January 7, 2020
Published in Issue Year 2020

Cite

APA Abuosba, E., & Ghanem, M. (2020). EM-HERMITE RINGS. International Electronic Journal of Algebra, 27(27), 88-101. https://doi.org/10.24330/ieja.662967
AMA Abuosba E, Ghanem M. EM-HERMITE RINGS. IEJA. January 2020;27(27):88-101. doi:10.24330/ieja.662967
Chicago Abuosba, Emad, and Manal Ghanem. “EM-HERMITE RINGS”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 88-101. https://doi.org/10.24330/ieja.662967.
EndNote Abuosba E, Ghanem M (January 1, 2020) EM-HERMITE RINGS. International Electronic Journal of Algebra 27 27 88–101.
IEEE E. Abuosba and M. Ghanem, “EM-HERMITE RINGS”, IEJA, vol. 27, no. 27, pp. 88–101, 2020, doi: 10.24330/ieja.662967.
ISNAD Abuosba, Emad - Ghanem, Manal. “EM-HERMITE RINGS”. International Electronic Journal of Algebra 27/27 (January 2020), 88-101. https://doi.org/10.24330/ieja.662967.
JAMA Abuosba E, Ghanem M. EM-HERMITE RINGS. IEJA. 2020;27:88–101.
MLA Abuosba, Emad and Manal Ghanem. “EM-HERMITE RINGS”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 88-101, doi:10.24330/ieja.662967.
Vancouver Abuosba E, Ghanem M. EM-HERMITE RINGS. IEJA. 2020;27(27):88-101.