Non-associative Ore extensions

TL;DR

This paper introduces non-associative Ore extensions, analyzes their ideal structure, and generalizes classical results on differential polynomial rings to the non-associative setting, including simplicity criteria and applications.

Contribution

It defines non-associative Ore extensions and characterizes their ideal structure, extending classical associative results to non-associative rings and exploring simplicity conditions.

Findings

Ideals are generated by monic polynomials in the center of the ring.

The center of the differential polynomial ring is explicitly described.

Conditions for simplicity of the non-associative differential polynomial rings are established.

Abstract

We introduce non-associative Ore extensions, $S = R[X ; \sigma , \delta]$, for any non-associative unital ring $R$ and any additive maps $\sigma,\delta : R \rightarrow R$ satisfying $\sigma(1)=1$ and $\delta(1)=0$. In the special case when $\delta$ is either left or right $R_{\delta}$-linear, where $R_{\delta} = \ker(\delta)$, and $R$ is $\delta$-simple, i.e. $\{ 0 \}$ and $R$ are the only $\delta$-invariant ideals of $R$, we determine the ideal structure of the non-associative differential polynomial ring $D = R[X ; \mathrm{id}_R , \delta]$. Namely, in that case, we show that all ideals of $D$ are generated by monic polynomials in the center $Z(D)$ of $D$. We also show that $Z(D) = R_{\delta}[p]$ for a monic $p \in R_{\delta}[X]$, unique up to addition of elements from $Z(R)_{\delta}$. Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of $D$ to show that $D$ is simple if and only if $R$ is $\delta$-simple and $Z(D)$ equals the field $R_{\delta} \cap Z(R)$. This provides us with a non-associative generalization of a result by \"{O}inert, Richter, and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of $D$ in the cases when the characteristic of the field $R_{\delta} \cap Z(R)$ is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.