ON PRIMARY AND REDUCED VARIETIES OF MONOASSOCIATIVE ALGEBRAS

Abstract

The notions of solvability, completeness (divisibility), primarity, reducibility, and purity play an important role in group theory. It turns out that there is another approach to these notions which leans on the theory of varieties of groups. This circumstance makes it possible to define analogs of these notions for arbitrary algebras. As far as solvability is concerned, this notion was introduced in [1] (also see [2]), wherein many problems were posed. Developing these ideas in [3], the author introduced the notions of primary, complete, and reduced algebras as well as that of pure subalgebra and some other notions. Some related results for modules and linear algebras were also mentioned in [3]. In the present article we characterize primary varieties of arbitrary monoassociative algebras over an arbitrary associative and commutative unital ring and characterize reduced varieties with certain properties for monoassociative algebras over Dedekind rings. In particular, these characterizations cover the cases of associative, alternative, Lie-type, and Jordan algebras (the corresponding results for rings were announced in [4]).