Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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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)
  • Received : 2009.09.30
  • Accepted : 2009.11.18
  • Published : 2009.12.25
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Abstract

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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References

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