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Invertibility and Dedekind finiteness in structural matrix rings

An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, Subrings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068–1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subrings M n (θ, R) of the full n × n matrix ring M n (R) over a ring R, which are closed with respect to taking inverses. It turns out that M n (θ, R) is closed with respect to taking inverses in M n (R) if all the equivalence classes with respect to θ ∩ θ−1, except possibly one, are of a size less than or equal to p (say) and M p (R) is Dedekind finite. Another purpose of this article is to show that M n (θ, R) is Dedekind finite if and only if M m (R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ−1). This provides a positive result for the inheritance of Dedekind finiteness by a matrix ring (albeit not a full matrix ring) from a smaller (full) matrix ring.