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,\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.Comment: 16 page

Non-unital Ore extensions

In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta]$, under the hypothesis that $R$ is $s$-unital and $\ker(\delta)$ contains a nonzero idempotent. This result generalizes a result by \"Oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings

Hilbert's basis theorem and simplicity for non-associative skew Laurent polynomial rings and related rings

We introduce non-associative skew Laurent polynomial rings over unital, non-associative rings. We prove simplicity results and a Hilbert's basis theorem for these. We also prove several versions of Hilbert's basis theorem for non-associative Ore extensions and non-associative generalizations of skew power series rings and skew Laurent series rings. For non-associative skew Laurent polynomial rings, we show that both a left and a right version of Hilbert's basis theorem hold. For non-associative Ore extensions, we show that a right version holds, but give a counterexample to a left version.Comment: 16 pages; added an example; proved left version of main theore

Hilbert's basis theorem for non-associative and hom-associative Ore extensions

We prove a hom-associative version of Hilbert's basis theorem, which includes as special cases both a non-associative version and the classical associative Hilbert's basis theorem for Ore extensions. Along the way, we develop hom-module theory, including the introduction of corresponding isomorphism theorems and a notion of being hom-noetherian. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.Comment: 18 pages; simplified the proof of Proposition 12; changed the title, abstract and introduction, added an example and a corollary together with some minor changes; corrected some minor typos in reference list; major revision including new title, shorter proofs and four new examples; added an example and updated an example, adjusted the languag