호남수학학술지 (Honam Mathematical Journal)

  • 제31권4호
  • /
  • Pages.515-527
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  • 2009
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  • 1225-293X(pISSN)
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  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
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MATRIX RINGS AND ITS TOTAL RINGS OF FRACTIONS

  • Lee, Sang-Cheol (Department of Mathematics Education Chonbuk National University, Department of Mathematics The University of Colorado)
  • 투고 : 2009.09.30
  • 심사 : 2009.11.18
  • 발행 : 2009.12.25
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초록

Let R be a commutative ring with identity. Then we prove $M_n(R)=GL_n(R)$ ${\cup}${$A{\in}M_n(R)\;{\mid}\;detA{\neq}0$ and det $A{\neq}U(R)$}${\cup}Z(M-n(R))$ where U(R) denotes the set of all units of R. In particular, it will be proved that the full matrix ring $M_n(F)$ over a field F is the disjoint union of the general linear group $GL_n(F)$ of degree n over the field F and the set $Z(M_n(F))$ of all zero-divisors of $M_n(F)$. Using the result and universal mapping property we prove that $M_n(F)$ is its total ring of fractions.

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참고문헌

  1. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969.
  2. D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer-Verlag New York, Inc., 1995.
  3. S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, Fourth Edition , Pearson Education, Inc., 2003.
  4. B. Kolman, Elementary Linear Algebra, Sixth Edition, Prentice-Hall, Inc., 1996.
  5. I. Kaplansky, Commutative Rings, The University of Chicago Press, Revised Edition, 1974.
  6. J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, Inc., 1988.