Patrick N. Stewart

ON THE GROENEWALD-HEYMAN STRONGLY PRIME RADICAL

Strongly prime rings were introduced by Handelman and Lawrence [6], and in a recent paper [5] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we continue this investigation. Section 1 provides some alternative characterizations of the radical and in section 2 we discuss general properties of the radical and compare it with other well-known radicals. Finally, combinatorial results on polynomial identities are presented which, combined with our results in section 2. yield some new comnutativity theorems. All rings considered are associative, but do not necessarily have an identity. As usual, I Δ A means that I is an ideal of the ring A. The notation and (xl,x2,…) will stand for the subring and ideal, respectively, generated by the elements x1,x2,…. The rignt annihirator of a subset S of a ring A will be denoted by annA(S). This work was supported in part by NSERC grants A-8775 and A-8789. and was completed while the first and ...

Radical Theory for Granded 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.

Injectives for ring isomorphisms with accessible images, ii

Those injectives for the category of associative rings and homomorphisms with accessible images which are indecomposable, and have torsion-free reduced additive groups, are characterized as principal ideal domains with every proper homomorphic image isomorphic to some Zn . This effectively completes the description of the injectives in the category referred to. Such rings can also be embedded in the p-adic integers whenever pA ≠ A.

TIGHT EXTENSIONS AND Λ-CLASSES

Abstract Certain classes of associative rings are characterized as finite subdirect products of specialized prime rings. These characterizations depend on results about Λ-classes and tight extensions, concepts that were fist introduced by Olson, Le Row and Hey man.

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentAe and essential ideals of the smash productA#G*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.