The $\mathbf{\aleph}_1$-Categoricity of Strictly Upper Triangular Matrix Rings Over Algebraically Closed Fields

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 ⊨ φ if and only if R φ σ(φ). Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 - categorical .