Abstract

Let n ≥ 3. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable. Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ 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 $R \vDash \sigma$. Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 -categorical.