Research Article
BibTex RIS Cite
Year 2020, , 13 - 42, 07.01.2020
https://doi.org/10.24330/ieja.662946

Abstract

References

  • G. Agnarsson, On a class of presentations of matrix algebras, Comm. Algebra, 24(14) (1996), 4331-4338.
  • G. Agnarsson, S. A. Amitsur and J. C. Robson, Recognition of matrix rings II, Israel J. Math., 96(part A) (1996), 1-13.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
  • G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • A. W. Chatters, Matrices, idealisers and integer quaternions, J. Algebra, 150(1) (1992), 45-56.
  • A. W. Chatters, Nonisomorphic rings with isomorphic matrix rings, Proc. Edinburgh Math. Soc. (Ser. 2), 36(2) (1993), 339-348.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999.
  • T. Y. Lam and A. Leroy, Recognition and computations of matrix rings, Israel J. Math., 96(part B) (1996), 379-397.
  • J. C. Robson, Recognition of matrix rings, Comm. Algebra, 19(7) (1991), 2113- 2124.
  • L. H. Rowen, Ring Theory: Student Edition, Academic Press, Inc., Boston, MA, 1991.
  • S. P. Smith, An example of a ring Morita equivalent to the Weyl algebra A1, J. Algebra, 73(2) (1981), 552-555.

ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Year 2020, , 13 - 42, 07.01.2020
https://doi.org/10.24330/ieja.662946

Abstract

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $R\cong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $\gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2(\field)$ when $\field$ is a base field $\rats$ or $\ints_p$ for a prime number $p$.

References

  • G. Agnarsson, On a class of presentations of matrix algebras, Comm. Algebra, 24(14) (1996), 4331-4338.
  • G. Agnarsson, S. A. Amitsur and J. C. Robson, Recognition of matrix rings II, Israel J. Math., 96(part A) (1996), 1-13.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
  • G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • A. W. Chatters, Matrices, idealisers and integer quaternions, J. Algebra, 150(1) (1992), 45-56.
  • A. W. Chatters, Nonisomorphic rings with isomorphic matrix rings, Proc. Edinburgh Math. Soc. (Ser. 2), 36(2) (1993), 339-348.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999.
  • T. Y. Lam and A. Leroy, Recognition and computations of matrix rings, Israel J. Math., 96(part B) (1996), 379-397.
  • J. C. Robson, Recognition of matrix rings, Comm. Algebra, 19(7) (1991), 2113- 2124.
  • L. H. Rowen, Ring Theory: Student Edition, Academic Press, Inc., Boston, MA, 1991.
  • S. P. Smith, An example of a ring Morita equivalent to the Weyl algebra A1, J. Algebra, 73(2) (1981), 552-555.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Geir Agnarsson This is me

Samuel S. Mendelson This is me

Publication Date January 7, 2020
Published in Issue Year 2020

Cite

APA Agnarsson, G., & Mendelson, S. S. (2020). ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS. International Electronic Journal of Algebra, 27(27), 13-42. https://doi.org/10.24330/ieja.662946
AMA Agnarsson G, Mendelson SS. ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS. IEJA. January 2020;27(27):13-42. doi:10.24330/ieja.662946
Chicago Agnarsson, Geir, and Samuel S. Mendelson. “ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 13-42. https://doi.org/10.24330/ieja.662946.
EndNote Agnarsson G, Mendelson SS (January 1, 2020) ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS. International Electronic Journal of Algebra 27 27 13–42.
IEEE G. Agnarsson and S. S. Mendelson, “ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS”, IEJA, vol. 27, no. 27, pp. 13–42, 2020, doi: 10.24330/ieja.662946.
ISNAD Agnarsson, Geir - Mendelson, Samuel S. “ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS”. International Electronic Journal of Algebra 27/27 (January 2020), 13-42. https://doi.org/10.24330/ieja.662946.
JAMA Agnarsson G, Mendelson SS. ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS. IEJA. 2020;27:13–42.
MLA Agnarsson, Geir and Samuel S. Mendelson. “ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 13-42, doi:10.24330/ieja.662946.
Vancouver Agnarsson G, Mendelson SS. ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS. IEJA. 2020;27(27):13-42.