Derivations in prime rings

LetR be a prime ring andD a nonzero derivation ofR. If one of the four conditions holds inR, thenR is commutative:

1. (i)
   X ²D(X)?D(X)X²∈Z(R), CharR≠2;     
   
   

Graded prime Goldie rings

Prime Goldie rings graded by a group are studied. It is proved that there exists a completely gr-reducible graded ring of fractions in the case of a grading group with a finite commutator subgroup (which enhances the results of Goodearl–Stafford obtained for Abelian groups). An example where such ring does not exist is constructed (a counterexample for a semi-prime ring was already known).     

Group rings which are G-Dedekind prime rings

Let R be a commutative Noetherian domain and A be a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and A when the group ring R[A] is a G-Dedekind prime ring.     

On rings whose prime ideals are completely prime

We investigate in this paper rings whose proper ideals are 2-primal. We concentrate on the connections between this condition and related concepts to this condition, and several kinds of π-regularities of rings which satisfy this condition. Moreover, we add counterexamples to the situations that occur naturally in the process of this note.     

Monomial conditions on prime rings

The study of pivotal monomials (and related conditions) is continued and extended, with the aim of studying carefully a situation generalizing Martindale's theory of prime rings with generalized polynomial identity. This is used to describe various classes of rings in terms of simple elementary sentences. The focus is on prime “Johnson” rings, which play a crucial role in our characterizations. It turns out that these rings can be characterized in terms of generalized pivotal monomials, thereby yielding a theory similar to that of [17]. An erratum to this article is available at .     

Crossed products over prime rings

In this paper we obtain necessary and sufficient conditions for the crossed productR *G to be prime or semiprime under the assumption thatR is prime. The main techniques used are the Δ-methods which reduce these questions to the finite normal subgroups ofG and a study of theX-inner automorphisms ofR which enables us to handle these finite groups. In particular we show thatR *G is semiprime ifR has characteristic 0. Furthermore, ifR has characteristicp>0, thenR *G is semiprime if and only ifR *P is semiprime for all elementary abelianp-subgroupsP of Δ⁺(G) ∩G _inn.     

Differential polynomial rings which are generalized Asano prime rings

Let R[x; δ] be a differential polynomial ring over a prime Goldie ring R in an indeterminate x, where δ is a derivation of R. In this paper, we describe explicitly the group of δ-stable v-R-ideals and using this results, we show that R[x; δ] is a generalized Asano prime ring if and only if R is a δ-generalized Asano prime ring.     

On strongly prime rings and ideals

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.