Frobenius Singularities of Algebraic Sets of Matrices

When one studies certain rings, it is natural to classify them according to certain properties. This project focuses on the study of properties of commutative rings associated with algebraic sets. In particular, we consider the algebraic set of pairs of square matrices whose commutator has a zero diagonal. We prove that it is irreducible and F-regular for matrices of all sizes and when the matrix entries are from a eld of positive prime characteristic. In addition, we provide a proof of its F-purity and nd a system of parameters on it. Moreover, we state several conjectures associated to this topic