Let $n \geq 3$. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular $n \times n$ matrix rings over fields is finitely axiomatizable. Theorem. If $R$ is a strictly upper triangular $n \times n$ matrix ring over a field $K$, then there is a recursive map $\sigma$ from sentences in the language of rings with constants for $K$ into sentences in the language of rings with constants for $R$ such that $K \vDash \varphi$ if and only if $R \vDash \sigma(\varphi)$. Theorem. The theory of a strictly upper triangular $n \times n$ matrix ring over an algebraically closed field is $\aleph_1$-categorical.