A Model Theoretic Perspective on Matrix Rings

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings Mₙ(K) in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This
is used together with invariant theory to prove quantifier elimination when K is an intersection of real closed fields. On the other hand, it is shown that finding a natural definable expansion with quantifier elimination of the theory
of Mₙ(ℂ) is closely related to the infamous simultaneous conjugacy problem in matrix theory. Finally, for various natural structures describing dimension-free matrices it is shown that no such elimination results can hold by establishing undecidability results.