ON BASE RADICAL OPERATORS FOR CLASSES OF FINITE ASSOCIATIVE RINGS

In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.

On Rings Whose Strongly Prime Ideals Are Completely Prime

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.

Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

On questions of B. J. Gardner and A. D. Sands

An example of two disjoint special classes whose upper radicals coincide is presented. It is shown that the left hereditary subradical of the hereditary idempotent radical is right hereditary. An example of a hereditary and principally left hereditary radical which is not left hereditary is constructed.

On homogeneous radicals of semigroup rings of commutative semigroups

SynopsisAll Archimedean commutative semigroups S are described such that every S-homogeneous hereditary radical is S-normal. It is shown that this result is in a sense unimprovable.

On semigroup algebras of cancellative commutative semigroups

SynopsisA cancellative commutative semigroup s and a hereditary radical ρ are constructed such that ρ is S-homogeneous but not S-normal. This answers a question which arose in the literature.

Radicals and subdirect decompositions of Ω-groups

Starting with a class ℳ of Ω-groups, necessary and sufficient conditions on ℳ are given to ensure that the corresponding Hoehnke radical ρ (determined by the subdirect closure of ℳ as semisimple class) is a radical in the sense of Kurosh and Amitsur; has a hereditary semisimple class; satisfies the ADS-property; has a hereditary radical class or satisfies ρN ∩ I ⊆ ρI and lastly, have both a hereditary radical and semisimple class or satisfies ρN ∩ I = ρI.

Radicals and one-sided ideals: an addendum

SynopsisIn the paper referred to in the title [2] an open question was raised concerning the equality of the largest left hereditary radical and the largest right hereditary radical contained in each of certain radicals. In this addendum an affirmative answer is provided to this question.

On Sands' questions concerning strong and hereditary radicals

[7] Sands raised the following questions:(1) Must a hereditary radical which is right strong be left strong?(2) Must a right hereditary radical be left hereditary?(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?Negative answers to questions (1) and (2) were given by Beidar [1].In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).