Veldsman’s classes of associative rings

Abstract

The relation of being an ideal (left ideal, right ideal) of a ring is not transitive. Rings for which the transitivity does hold are called filial (left filial, right filial). In symbols, these are rings R such that if AαB and BαR then AαR, where α is the respective relation. In [24] Veldsman introduced for each triple (α, β, γ) of these relations the class of rings R which satisfy: if AαB and BβR then AγR. He found some relationships among so defined classes and characterized rings in each of them. Our aim in this paper is to study the structure of rings in these classes. We observe that one can select three from among them, which are fundamental in the sense that results obtained for them together with those known for left filial rings, give structure theorems for rings in all classes. Then we study the structure of rings in so selected three classes. Our main results concern the prime and generalized nil radicals of such rings and reduced rings in these classes. We also get many related results and ask several questions.