Model theory of strictly upper triangular matrix rings

Extract

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (α_{i, j}) is strictly upper triangular if α_{i, j} = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by S_n(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of S_n(F) for an algebraically closed field F is ℵ₁categorical. The first main result of this paper is that the rings S_n(F) and S_n(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].