On irreducible semigroups of matrices with traces in a subfield (Q1826830)

Let \(S\) be an irreducible subsemigroup of the matrix algebra \(M_n(K)\), where \(K\) is a field, and write \(\text{tr}(S)= \{\text{tr}(X): X\in S\}\). The author proves an interesting refinement of the classical Wedderburn-Artin theorem under the assumption that \(\{0\}\neq \text{tr}(S)\subseteq F\), where \(F\) is a subfield of \(K\). Theorem 2.9 asserts that, if \(r\) is the smallest nonzero rank of any element of \(S\), then \(r\) is a divisor of \(n\) and \(\text{Alg}_F(S)\) is similar in \(M_n(K)\) to an explicitly exhibited matrix algebra \(M_{n/r}({\mathcal D}_r)\), where \({\mathcal D}_r\) is an irreducible division \(F\)-algebra with \(\{0\}\neq \text{tr}({\mathcal D}_r)\subseteq F\). There are some added remarks on the applicability of the theorem and the special case where \(F\) is finite is discussed in some detail.