Lower radicals in nonassociative rings

Abstract

Let W be a universal class of (not necessarily associative) rings and let A ⊆ W. Kurosh has given in [6] a construction for LA, the lower radical class determined by A in W. Using this construction, Leavitt and Hoffmann have proved in [4] that if A is a hereditary class (if K ∈ A and I is an ideal of K, then I ∈ A), then LA is also hereditary. In this paper an alternate lower radical construction is given. As applications, a simple proof is given of the theorem of Leavitt and Hoffmann and a result of Yu-Lee Lee for alternative rings is extended to not necessarily associative rings.

A Note on the Upper Radicals of Seminearrings

“…We extend the results of [2,5,6,7,9,10,12] by using the above construction of upper radical for seminearring which is indeed provides an excellent and different approach to handle the many results of [2,5,6,7,9,10,12] in the frame work of seminearring.…”

“…Hereditary properties inherited by the lower radical generated by a class M have been considered in [2,5,6,7,9,10,12]. Here we consider the dual problem, namely strong properties which are inherited by the upper radical generated by a class M .…”

“…The upper radicals were investigated by (see [2,9,12]) for radical classes of rings. Here we are interested in generalizing several results from [2,5,6,7,9,10,12] in the framework of seminearring, which is quite different from the ring theoretical approach discussed in [2,5,6,7,9,10,12]. Throughout this paper N will denote an seminearrings and ω be the universal class of all seminearrings.…”

“…Lower radical classes for seminearrings can be constructed similar to the construction of lower radicals for rings (see [2,5,6,7,8,9,10,12,15,16]). First we include necessary preliminary, let ω be the universal class of all seminearrings and M be a sub-class of ω and let M 0 be the homomorphic closure of M in ω.…”

“…First we give a construction for the upper radical, dual to the construction of [10] for the lower radical. From this the theorem on the inheritance of the left strong property is deduced.…”