Maximal Commutative Subrings and Simplicity of Ore Extensions

Öinert, Johan; Richter, Johan; Silvestrov, Sergei

doi:10.1142/s0219498812501927

Cited by 14 publications

(18 citation statements)

References 38 publications

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“…cit. simplicity results were obtained for these more general types of differential polynomial rings R[x; id R , δ], thereby generalizing the results of [2], [16] and [24].…”

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confidence: 72%

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Simplicity of Ore monoid rings

2019

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Given a non-associative unital ring R, a monoid G and a set π of additive maps R → R, we introduce the Ore monoid ring R[π; G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.

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“…cit. simplicity results were obtained for these more general types of differential polynomial rings R[x; id R , δ], thereby generalizing the results of [2], [16] and [24].…”

supporting

confidence: 72%

“…The claim now follows immediately from Theorem 18 and Proposition 21(f). [24,Theorem 4.15] both to the case of several variables and to the non-associative situation. In fact, if R is associative, I is the finite set {1, .…”

Section: The Monoid N (I)mentioning

confidence: 99%

Simplicity of Ore monoid rings

2019

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“…In [12] Öinert, Richter and Silvestrov show that there are simplicity results for differential polynomial rings that are almost completely analogous to the skew group ring situation (for more details, see Section 3.7). Theorem 1.2 (Öinert, Richter and Silvestrov [12]). Suppose that B is an associative and unital ring and A = B[x; δ] is a differential polynomial ring.…”

Section: Introductionmentioning

confidence: 98%

Simple rings and degree maps

Nystedt

Öinert

2014

Journal of Algebra

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For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying AI ⊆ IA. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-negative integers satisfying the following two conditions (d1) if a ∈ A, then d(a) = 0 if and only if a = 0; (d2) there is a subset X of B generating B as a ring such that for each non-zero ideal I of A and each non-zero a ∈ I there is a non-zeroWe show that if the centralizer C of B in A is an A-simple ring, every intersection of C with an ideal of A is A-invariant, ACA = A and there is a degree map for A/B, then A is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings.

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“…Theorem 4 (Öinert, Richter and Silvestrov [22]). If R is associative and δ : R → R is a derivation, then D = R[X; id R , δ] is simple if and only if R is δ-simple and Z(D) is a field.…”

Section: Introductionmentioning

confidence: 98%

“…Special attention is also often paid to the case when R is commutative. However, in [22] Öinert, Richter and Silvestrov have shown the following simplicity result that holds for all associative differential polynomial rings regardless of characteristic.…”

Section: Introductionmentioning

confidence: 99%

Non-associative Ore extensions

2018

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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.

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Maximal Commutative Subrings and Simplicity of Ore Extensions

Cited by 14 publications

References 38 publications

Simplicity of Ore monoid rings

Simplicity of Ore monoid rings

Simple rings and degree maps

Non-associative Ore extensions

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