Maximal commutative subrings and simplicity of Ore extensions
Öinert, Johan; Richter, Johan; Silvestrov, Sergei (2013). Maximal commutative subrings and simplicity of Ore extensions. Journal of Algebra and Its Applications, 12, (4), 16 - 1250192
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Published
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English
Authors:
Öinert, Johan
;
Richter, Johan
;
Silvestrov, Sergei
Department:
Mathematics (Faculty of Engineering)
Non-commutative Geometry-lup-obsolete
Research Group:
Non-commutative Geometry-lup-obsolete
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_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.
Keywords:
Ore extension rings ;
maximal commutativity ;
ideals ;
simplicity
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