MATRIX RINGS AND ITS TOTAL RINGS OF FRACTIONS

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 4,  2009, pp.515-527
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.4.515
Title & Authors
MATRIX RINGS AND ITS TOTAL RINGS OF FRACTIONS
Lee, Sang-Cheol;

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