R. R. Andruszkiewicz

The Classification of Integral Domains in Which the Relation of Being an Ideal Is Transitive

Abstract A ring R is said to be filial when for every I, J, if I is an ideal of J and J is an ideal of R then I is an ideal of R. The complete classification and the method of construction of commutative filial domains is given.

The Classification of Commutative, Noetherian, Filial Rings with Identity

Filial rings are rings in which the relation of being an ideal is transitive. We continue the study of commutative filial rings started in [2, 3]. In particular, we give a complete description of the commutative, filial, noetherian rings with identity.

ON COMMUTATIVE REDUCED FILIAL RINGS

A ring in which every accessible subring is an ideal is called filial. We continue the study of commutative reduced filial rings started in [R. R. Andruszkiewicz and K. Pryszczepko, ‘A classification of commutative reduced filial rings’, Comm. Algebra to appear]. In particular we describe the Noetherian commutative reduced rings and construct nontrivial examples of commutative reduced filial rings without ideals which are domains.

A Classification of Commutative Reduced Filial Rings

A ring R is said to be filial when for every I, J, if I is an ideal of J and J is an ideal of R then I is an ideal of R. The classification of commutative reduced filial rings is given.

Some New Results for the Square Subgroup of an Abelian Group

The complete description of the square subgroup of a torsion abelian group and an elementary construction of a mixed abelian group (A, +, 0), such that the quotient group of A modulo the square subgroup □A is not a nil-group, are given (also for the associative case). Some A. M. Aghdams results concerning square subgroups for the associative case are proven. The relationship between square subgroups for both the cases of associative and general rings is partially investigated.

Kurosh's chains of associative rings

Let N be a homomorphically closed class of associative rings. Put N 1 = N l = N and, for ordinals a ≥ 2, define N α ( N α ) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in N β for some β N α } ({ N α }), the union of which is equal to the lower radical class IN (lower left strong radical class IsN ) determined by N . The chain { N α } is called Kuroshs chain of N . Sulinski, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ N k for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = N n +l ≠ N n . Some results concerning the termination of the chain { N α } were obtained in [2,4]. In this paper we present some classes N with N α = N α for all α Using this and Beidars example we prove that for every natural number n ≥ 1 there exists an N such that N α = N α for all α and N n ≠ N n+i = N n+2 . This in particular answers Question 6 of [4].

On Associative Ring Multiplication on Abelian Mixed Groups

An abelian group is called a mixed one if it is neither torsion nor torsion-free. It is to be proved that every mixed group can be provided with a nonzero associative ring structure. Our methods of proofs are straightforward and elementary.

THE CLASSIFICATION OF COMMUTATIVE TORSION FILIAL RINGS

The aim of this paper is to give a classification theorem for commutative torsion filial rings. 2010 Mathematics subject classification: primary 16D25; secondary 13C05, 13B02.

On additive groups of associative and commutative rings

Abstract The characterization of (A)CR-groups studied by Shalom Feigelstock [‘Additive groups of commutative rings’, Quaest. Math. 23 (2000), 241–245] is complemented. In particular, the theorem characterizing mixed ACR-groups is proved and numerous new examples of (A)CR-group are given. Abelian groups A such that any ring with the additive group A is associative and commutative are partially described. Some new results for torsion-free and mixed TI-groups are obtained.

ON THE STABILISATION OF ONE-SIDED KUROSH’S CHAINS

We construct an example showing that Kurosh’s construction of the lower strong radical in the class of associative rings may not terminate at any finite ordinal.