Maximal commutative subrings and simplicity of Ore extensions

Abstract: The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,\delta], is simple if and only if its center is a field and R is \delta-simple. When R is commutative we note that the centralizer of R in R[x;\sigma,\delta] is a maximal commutative subring containing R and, in the case when \sigma=id, we show that it intersects every non-zero ideal of R[x;id,\delta] non-trivially. Using this we show that if R is \delta-simple and maximal commutative in R[x;id,\delta], then R[x;id,\delta] is simple. We also show that under some conditions on R the converse holds.