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arXiv · 1111.1292

Maximal commutative subrings and simplicity of Ore extensions

Johan Öinert·Johan Richter·Sergei D. Silvestrov
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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,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] is simple. We also show that under some conditions on R the converse holds.

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Johan Öinert, Johan Richter, Sergei D. Silvestrov. 2012-08-30. Maximal commutative subrings and simplicity of Ore extensions. https://doi.org/10.1142/s0219498812501927

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