 Matrix and triangular localization over an Ore domain

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 $a{I}_r$, proving the displayed classical localizations. The triangular localization retains its strictly upper triangular nilpotent ideal; it is not a semisimple matrix ring.