A generalization of the lower radical type construction for strict and base radical classes

If R is a ring and n is an integer weMaydefine a ring Tn (R) on the same underlying additive abelian group by using the formula a * b = nab to define a new multiplication. Tn , is a functor on the category of associative rings. If C is a class of rings then, for each n, the class Cn , is defined to consist of all rings R such that Tn (R) is in C. If C is a radical class then each class Cn , is also a radical class. We consider the properties of the radical class C which are inherited by Cn , and relationships between these classes C n as n varies.

Some Pathology for Radicals in Non-Associative Near-Rings

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.

Hemirings are one of the Universal algebras generalizing an associative ring sharing the same properties as a ring except that the second binary operation is assumed to be a semigroup rather than a group. Many researchers made a detailed study on different aspects of hemirings. This paper introduces the notions of prime and semiprime intuitionistic fuzzy left k-ideals of hemirings. We also investigate some of their properties.

To the Abstract Theory of Radicals: A Contribution from Near-Rings

In this paper we propose a general setting in which to study the radical theory of group graded rings. If is a radical class of associative rings we consider two associated radical classes of graded rings which are denoted by G and ref. We show that if is special (respectively, normal), then both G and ref are graded special (respectively, graded normal). Also, we discuss a graded version of the ADS theorem and the termination of the Kurosh lower graded radical construction.