Maximality preserving constructions of maximal commutative subalgebras of matrix algebra (Q2880469)

Let \((R, m_R, k)\) be a local maximal commutative subalgebra of the matrix algebra \(M_n(k)\). The author proves that \(R\) is isomorphic to the algebra \(R^{ST}\) of all skew transpose matrices of elements of \(R\) under a maximality preserving map, and study some of the interesting properties related to \(R^{ST}\). In section 3, some conditions in which \(S=\{r \in R ~|~ r^{TS}=r \}\) becomes an algebra is stated, and, in section 4, a method to construct a local maximal commutative subalgebra \(R\) of \(M_n(k)\) with index of maximality \(i(m_R) =\dim (R) = n\) and such that \(r=r^{ST}\) for all \(r \in m_R\) is introduced.