Maximal commutative subrings and simplicity of Ore extensions

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_R,\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_R, we show that it intersects every non-zero ideal of R[x;id_R,\delta] non-trivially. Using this we show that if R is \delta-simple and maximal commutative in R[x;id_R,\delta], then R[x;id_R,\delta] is simple. We also show that under some conditions on R the converse holds.