Matrix rings over polynomial identity rings

Abstract

We prove that if $A$ is an algebra over a field with at least $k$ elements, and $A$ satisfies ${x^k} = 0$, then ${A_n}$, the ring of $n$-by-$n$ matrices over $A$, satisfies ${x^q} = 0$, where $q = k{n^2} + 1$. Theorem 1.3 generalizes this result to rings: If $A$ is a ring satisfying ${x^k} = 0$, then for all $n$, there exists $q$ such that ${A_n}$ satisfies ${x^q} = 0$. Definitions. A