Maximal commutative subrings and simplicity of Ore extensions

Details Maximal commutative subrings and simplicity of Ore extensions Maximal commutative subrings and simplicity of Ore extensions

2011-11-05

arxiv.org/abs/1111.1292v1

Mathematics

Rings and Algebras

21 pages

Scientific paper

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. In the case 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 coincides with its centralizer in R[x;id,\delta], then R[x;id,\delta] is simple. We also show that under some conditions on R the converse holds.

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