Non-associative Ore extensions

Abstract

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ : R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R δ -linear, where R δ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the non-associative differential polynomial ring D = R[X; idR, δ]. 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 δ [p] for a monic p ∈ R δ [X], unique up to addition of elements from Z(R) δ . 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 δ-simple and Z(D) equals the field R δ ∩ Z(R). This provides us with a non-associative generalization of a result by Öinert, 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 δ ∩ 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.

Simplicity of Ore monoid rings

“…Base case: n = 1. This has already been proven in [23,Proposition 3.7]. Induction step: Suppose that n > 1 and that the claim holds for all elements in N (I) such that the cardinality of the support of the element is less than n. Suppose that f = g + h, for some g, h ∈ N (I) , is chosen so that the cardinalities of the supports of g and h are less than n. From the induction hypothesis and Proposition 26, we now get that which holds for all a, b, c ∈ N.…”

“…In that case, R[G; π] is called a classical Ore monoid ring (classical differential monoid ring). It follows from the examples in [23] (for the case G = N) that the following inclusions hold:…”

“…The main objective of this article is to extend the simplicity results in [23] to Ore monoid rings. The secondary objective is to apply our extended results to particular cases of monoids to obtain new proofs of classical simplicity results for iterated Ore extensions as well as showing simplicity results for new Ore-like structures.…”

Hom-Associative Ore Extensions and Weak Unitalizations

“…A well-known fact about the associative Weyl algebras are that they are simple. This fact is also true in the case of the non-associative Weyl algebras introduced in [23], and it turns out that the hom-associative Weyl algebras have this property as well.…”

“…Non-associative Ore extensions on the other hand were first introduced in 2015 and in the unital case, by Nystedt,Öinert, and Richter [24] (see also [23] for an extension to monoid Ore extensions). In the present article, we generalize this construction to the non-unital case, as well as investigate when these non-unital, nonassociative Ore extensions are hom-associative.…”