Higher derivations of Ore extensions by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-skew derivations

2016

Journal of Algebra

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MSC: 16N60 16R50 16S40 16T15 16W20 16W22 16W25 16W25 Keywords: Prime ring Automorphism Skew derivation Identity Coalgebras Pointed coalgebras The complete ring of quotients Utumi quotient ringLet R be a prime ring with the extended centroid c. Suppose that R is acted by a pointed coalgebra with group-like elements acting as automorphisms of R. A generalized polynomial with variables acted by the coalgebra is called an identity if it vanishes on R. We prove the following:(1) If c is a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R with variables acted by Frobenius automorphisms.(2) If c is not a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R. With this, we extend Yanai's result [25] to "nonlinear identities". These are actually special instances of our Theorems 1 and 2 below respectively, which extend Kharchenko's theory of differential identities [14,15] to the context of expansion closed word sets introduced in [6].

Nonlinear identities with skew derivations

2016

Journal of Algebra

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Invariant polynomials of Ore extensions by $q$-skew derivations

Lee

Liu

2012

Proc. Amer. Math. Soc.

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Abstract. Let R be a prime ring with the symmetric Martindale quotient ring Q. Suppose that δ is a quasi-algebraic q-skew σ-derivation of R. For a minimal monic semi-invariant polynomial π(t) of Q[t; σ, δ], we show that π(t) is also invariant if char R = 0 and that either π(t) − c for some c ∈ Q or π(t) p is a minimal monic invariant polynomial if char R = p ≥ 2. As an application, we prove that any R-disjoint prime ideal of R [t; σ, δ] is the principal ideal p(t) for an irreducible monic invariant polynomial p(t) unless σ or δ is X-inner.

Smash products and differential identities

Trans. Amer. Math. Soc.

Tsai

2012

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Let U be the universal enveloping algebra of a Lie algebra and R a U-module algebra, where U is considered as a Hopf algebra canonically. We determine the centralizer of R in R#U with its associated graded algebra. We then apply this to the Ore extension R[X; φ], where φ : X → Der(R). With the help of PBW-bases, the following is proved for a prime ring R: Let Q be the symmetric Martindale quotient ring of R. For f i , g i ∈ Q[X; φ], i f i rg i = 0 for all r ∈ R iff i f i ⊗ g i = 0, where ⊗ is over the centralizer of R in Q[X; φ]. Finally, we deduce from this Kharchenko's theorem on differential identities.