= Matrix and triangular localization over an Ore domain
{title2=$Q^l_{\mathrm{cl}}(M_r(A))=M_r(K),\quad Q^l_{\mathrm{cl}}(T_r(A))=T_r(K)$}
Let $A$ be a domain satisfying both Ore conditions, with quotient <division ring> $K$. A matrix over $A$ is regular exactly when it is invertible over $K$: a kernel vector of a singular matrix can have its denominators cleared to give a nonzero annihilator over $A$. An upper triangular matrix is regular exactly when all diagonal entries are nonzero. Each finite list of entries of a matrix over $K$ has a common left denominator $a\in A\setminus\{0\}$. Thus it is a fraction with denominator $aI_r$, proving the displayed classical localizations. The triangular localization retains its strictly upper triangular nilpotent ideal; it is not a semisimple matrix ring.
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