Simplicity of Ore monoid rings

Nystedt, Patrik; Öinert, Johan; Richter, Johan

doi:10.1016/j.jalgebra.2019.04.003

Cited by 6 publications

(3 citation statements)

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“…Ore extensions were first introduced under the name non-commutative polynomial rings by Ore [14]. Non-associative Ore extensions were introduced by Nystedt, Öinert, and Richter in the unital case [12] (see also [13] for a further extension to monoid Ore extensions). The construction was later generalized to non-unital, hom-associative Ore extensions by the authors of the present article and Silvestrov [3].…”

Section: Introductionmentioning

confidence: 99%

Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions

Bäck

Richter

2022

Algebr Represent Theor

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Section: Introductionmentioning

confidence: 99%

Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions

Bäck

Richter

2022

Algebr Represent Theor

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Hom-Associative Ore Extensions and Weak Unitalizations

Bäck¹,

Richter²,

Silvestrov³

2018

International Electronic Journal of Algebra

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We introduce hom-associative Ore extensions as non-unital, nonassociative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative homassociative algebra into a multiplicative, weakly unital hom-associative algebra, which we call a weak unitalization.2010 Mathematics Subject Classification. 17A30, 17A01. Key words and phrases. hom-associative Ore extensions, hom-associative Weyl algebras, homassociative algebras. Definition 2.1 (Hom-associative algebra). A hom-associative algebra over an associative, commutative, and unital ring R, is a triple (M, ·, α) consisting of an R-module M , a binary operation · : M × M → M linear over R in both arguments, and an R-linear map α : M → M satisfying, for all a, b, c ∈ M ,Since α twists the associativity, we will refer to it as the twisting map, and unless otherwise stated, it is understood that α without any further reference will always denote the twisting map of a hom-associative algebra.Remark 2.2. A hom-associative algebra over R is in particular a non-unital, nonassociative R-algebra, and in case α is the identity map, a non-unital, associative R-algebra.Furthermore, if the twisting map α is also multiplicative, i.e. if α(a · b) = α(a) · α(b) for all elements a and b in the algebra, then we say that the homassociative algebra is multiplicative.Definition 2.3 (Morphism of hom-associative algebras). A morphism between two hom-associative algebras A and A ′ with twisting maps α and α ′ respectively, is an algebra homomorphism f : A → A ′ such that f • α = α ′ • f . If f is also bijective, the two are isomorphic, written A ∼ = A ′ .

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“…Nystedt, Öinert and Richter [14,15] generalized the construction of Ore extensions to obtain non-associative Ore extensions, and the even broader class of Ore monoid rings, and generalized the conditions on simplicity from [16].…”

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Non-unital Ore extensions

Lundström¹,

Öinert²,

Richter³

2023

Colloq. Math.

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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.

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Simplicity of Ore monoid rings

Cited by 6 publications

References 25 publications

Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions

Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions

Hom-Associative Ore Extensions and Weak Unitalizations

Non-unital Ore extensions

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