Non-unital Ore extensions

Abstract

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; δ], under the hypothesis that R is s-unital and ker(δ) contains a non-zero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings. Introduction.In this article all rings will be associative, but not necessarily unital. Ore extensions are unital rings that were introduced by Ore [17] under the name of non-commutative polynomial rings. Our aim here is to study a non-unital generalization of Ore extensions.Let R be a unital ring, let σ : R → R be a unital ring endomorphism (not necessarily injective), and let δ : R → R be a σ-derivation, i.e. δ is an additive map satisfying δ(rs) = σ(r)δ(s) + δ(r)s for all r, s ∈ R. The Ore extension R[x; σ, δ] is defined as the ring generated by R and by an element x / ∈ R such that 1, x, x 2 , . . . form a basis for R[x; σ, δ] as a left R-module and all r ∈ R satisfySuch a ring always exists and is unique up to isomorphism (see [9]). Sinceand hence 1 R will be a multiplicative identity element for R[x; σ, δ] as well.The Ore extensions play an important role when investigating cyclic algebras, enveloping rings of solvable Lie algebras, and various types of graded rings such as group rings and crossed products (see e.g.