Jerzy Matczuk

The extended centroid and X-inner automorphisms of Ore extensions

For the Ore extension R[t; S, D], where R is a prime ring, we determine the center, the extended centroid, and the X-inner automorphisms. The results depend on the structure of ideals of R[t; S, D].

Goldie rank of ore extensions

This paper is concerned with the problem when an Ore extension R[t, S, D] has the same Goldie rank as the coefficient ring R. It is shown, in particular, that this is the case when either R is non-singular or D is a q-quantized s-derivation satisfying some finiteness conditions.

On Induced Modules Over Ore Extensions

Abstract Let R be a ring and S = R[x;σ, δ] its Ore extension. For an R-module M R we investigate the uniform dimension and associated primes of the induced S-module M ⊗ R S.

Quasi-duo skew polynomial rings

A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.

A Characterization of σ -Rigid Rings

Abstract Let σ be an endomorphism of a ring R. It is shown that R is σ-rigid iff σ is injective, R is reduced and σ-skew Armendariz. This gives a positive answer to the question posed in Hong et al. [Hong, C. Y., Kim, N. K., Kwak, T. K. (2003). On skew Armendariz rings. Comm. Algebra 31(1):2511–2528].

Primitivity of skew polynomial and skew laurent polynomial rings

Let R be a noetherian P.I. ring and S an automorphism of R. Necessary and sufficient conditions for the primitivity of the skew Laurent polynomial ring R[t;t-1S] and the skew polynomial ring R[t,S] are given.

On the Gelfand-Kirillov Dimension of Normal Localizations and Twisted Polynomial Rings

It is well known that, if S is a central subset consisting of regular elements of an associative algebra A over a field K, then the algebras A and AS −1 have the same Gelfand-Kirillov dimension. This also holds if S is a commutative set of regular elements determining locally nilpotent inner derivations of A, cf. [2], Chapter 4. Some other positive results are concerned with the localizations of the enveloping algebras of finite dimensional Lie algebras over a field of characteristic zero, [1]. On the other hand, there are known examples showing that in general the Gelfand-Kirillov dimension of a localization may be very far from that of the original algebra, [2], Chapter 4.

Artinian property of constants of algebraicq-skew derivations

Let δ denote aq-skew σ-derivation of an algebraR andR(δ)={r εR│δ(r)=0} stand for the subalgebra of invariants. We prove thatR(δ) is left artinian iffR is left artinian providedR is semiprime and the action of δ onR is algebraic.

On the composition of derivations

Posner ([9]) has shown that for any prime ringR of characteristic different from 2 the composition of any two non-zero derivations is not a derivation. On the other hand, it is well known ([4]) that if charR=n for a prime numbern andd is a derivation ofR, thendn is also a derivation.Our main objective is to extend the above mentioned result of Posner in the case of commutative domains, and to apply this results to the investigation of connections either between derivations and a center, or between derivations and a generalized centroid of a prime ring. For this purpose, we are first going to introduce a method of notation for the composition of derivations which, we hope, will also be useful in other situations.

On Algebraic Derivations of Prime Rings

It is well known that if R is a ring of characteristic p>0 and d is a derivation of R, then dp is also a derivation. On the other hand, for a prime ring R, powers, less than char R, of inner derivations which are inner derivations were investigated in [3]. It appeared in particular that elements which determined such derivations have to be algebraic and the power of a derivation is not often a derivation.