A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).