Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

SPECIAL RADICALS FOR NEAR-RING

The concept of a special radical for near-rings has been treated in several nonequivalent, but related, ways in the recent literature. We use the version due to K. Kaarli to establish that various prime radicals and the nil radical are special radicals on the class A of all near-rings which satisfy an extended version of the Andrunakievich Lemma. Since A includes all d.g. near-rings—and much more—these results significantly extend results previously obtained by Kaarli and by Groenewald. We also obtain special radical results for the Jacobson type radicals 30 and 3 1 , albeit on less extensive classes. Examples are given which illustrate and delimit the theory developed.

Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ? N?.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 1-11

A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

Proper Fuzzification of Prime Ideals of a Hemiring

Prime fuzzy ideals, prime fuzzyk-ideals, and prime fuzzyh-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp.,k-ideal,h-ideal) is contained in a prime fuzzy ideal (resp.,k-ideal,h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzyk-ideal (resp., anh-ideal) is proved to be a fuzzyk-ideal (resp.,h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.

Isomorphism Characterization of Divisible Groups in Modular Abelian Group Rings

Suppose G is an abelian group with a p-subgroup H and R is a commutative unitary ring of prime characteristic p with trivial nil-radical. We give a complete description up to isomorphism of the maximal divisible subgroups of 1 + I(RG;H) and (1 + I(RG;H))=H, respectively, where I(RG;H) denotes the relative augmentation ideal of the group algebra RG with respect to H. This paper terminates a series of works by the author on the topic, first of which are [Danchev, Rad. Mat. 13: 23–32, 2004] and [Danchev, Bull. Georgian Acad. Sci. 174: 238–242, 2006].