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.     

Semirings which are unions of rings

Sernirings which are a disjoint union of rings form a variety S which contains the variety of all rings and the variety of all idempotent sernirings, and in particular, the variety of distributive lattices. Various structure theorems are established which bring insight into the structure of the lattice of subvarieties of S.``     

RRF rings which are not LRF

Define a ringA to be RRF (respectively LRF) if every right (respectively left)A-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF.     

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.     

Associative rings the radicals over which are subrings

The paper studies the structure of algebras which are radicals over PI-subalgebras. In particular, a theorem is proven to the effect that an algebra without nil-ideals which is a radical over a right Pi-ideal is a Pi-algebra.Translated from Matematicheskie Zametki, Vol.11, No.1, pp. 33–40, January, 1972.In conclusion, the author wishes to thank L. A. Bokut for help with and interest in this paper.     

Jacobson radical of skew polynomial rings and skew group rings

LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R _σ[x])={Σ_ir_i x ⁱ:r₀∈I∩J(R]), r _i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R _Σ[x])|s= (ii)J(R _σ<x>)=(J(R _σ<x>)∩R)_σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.     

Jacobson rings and rings strongly graded by an abelian group

LetR be ring strongly graded by an abelian groupG of finite torsion-free rank. Lete be the identity ofG, andR _e the component of degreee ofR. AssumeR _e is a Jacobson ring. We prove that graded subrings ofR are again Jacobson rings if eitherR _e is a left Noetherian ring orR is a group ring. In particular we generalise Goldie and Michlers’s result on Jacobson polycyclic group rings, and Gilmer’s result on Jacobson commutative semigroup rings of finite torsion-free rank.     

Linearly ordered rings which are not o-epimorphic images of ordered free rings

A proof is given that not every linearly ordered associative (associative-commutative) ring is the o-image of a free associative (associative-commutative) ring for some ordering of the latter. There are also nilpotent linearly ordered rings which are not o-epimorphic images of free associative or free associative-commutative n-nilpotent rings for n 4, no matter what ordering is used for the latter.Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 693–697, June, 1971.     

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by J_R, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 ? xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S_n. In this paper, we investigate the genus number of the compact Riemann surface in which J_R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that J_R is toroidal.