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Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X; S, D] is ν if char R = 0, and is p k ν for some k 0 if char R = p 2, where ν is the least integer ν 1 such that S ν D S −ν − D is X-inner. A similar result holds for cvpolynomials. These are done by introducing the new notion of kbasic polynomials for each integer k 0, which enable us to analyze semi-invariant polynomials inductively.Let σ , τ be automorphisms of a ring R.We call a (τ , σ )-derivation outer if it is not of this form. Let 1 denote the identity automorphism x → x. We call (1, σ )-derivations simply σ -derivations. We also write ad (1,σ ) (b) as ad σ (b). 2465 Let S be an automorphism of R and D an S-derivation of R. The Ore extension of R by the S-derivation D, denoted by R[ X; S, D], is the ring freely generated by the ring R adjoined by an indeterminate X subjected to the commuting rule Xr = S(r)X + D(r) for r ∈ R.Ore extensions, since their discovery in [20], have been very important in constructing interesting mathematical objects, see for example [1,2,[4][5][6][7][8][9][10][11][12][13][15][16][17][18][19][21][22][23][24][25]. In modern algebras, this elementary construction is important in particular due to the fact that the quantum Borel subalgebra in DrinfeldJimbo quantization is an iterated Ore extension, see for example [4].Throughout, R is a prime ring. To investigate R[ X; S, D], we have to work in the symmetric Martindale quotient ring of R, which we denote by Q . (See [3] for the definition and basic properties of Q .)The automorphism S and the S-derivation D can be uniquely extended to Q . We form Q [X; S, D] analogously. Elements of Q [X; S, D] are called skew polynomials or merely polynomials for short.Any skew polynomial f can be written uniquely in the form f = i a i X i , where a i ∈ Q vanish for all but finitely many i. We call a i the coefficient of X i in f . The degree of a nonzero polynomial f , denoted by deg f , is the largest integer m 0 with a m = 0 and the corresponding a m is called the leading coefficient of f . We call f monic if the leading coefficient a m is equal to 1. We also postulate that the zero polynomial has the degree −∞. So deg f g deg f + deg g for any f , g. The equality holds, in particular, if one of f , g is zero or has an invertible leading coefficient. It is convenient to have the following notations:= { f ∈ P: deg f < m} for integer m 0. Clearly, P(m) forms a (Q , Q )-bimodule. Let A be the group of automorphisms of Q . For σ , τ ∈ A, let L σ ,τ be the set of (σ , τ )-derivations and L i σ ,τ the set of all inner (σ , τ )-derivations. If σ = 1 then write

Abstract. Let R be a ring, X a sequence of noncommuting indeterminates

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.