The freeness theorem in some varieties of linear algebras and rings

In this paper we study properties of free and near-free linear $\Omega$-algebras (over a field) lying in a variety $\mathfrak{M}_P$ given by permutational identities. Examples of sub-identities in the case of ordinary linear algebras (with a single binary operation) are the commutative and anticommutative laws. The identities, studied by Polin in [4] are also special cases of identities of this kind. The auxiliary results derived in the first two sections yield a method of proof for varieties $\mathfrak{M}_P$ of the freeness theorem analoguos to the Dehn-Magnus theorem for groups [7], Zhukov's theorem for non-associative algebras [2], and Shirshov's theorems for commutative and anticommutative algebras [5] and Lie algebras [6]. Zhukov' s theorem [2] and Shirshov's theorem [5] are special cases of our proposition. We note that although generally speaking a subalgebra of a free algebra in $\mathfrak{M}_P$ need not be free in the variety, the freeness theorem is always true for such varieties. It is known that for non-associative rings, in contrast to the case of linear algebras, the theorem on subrings of a free ring is false in the most general case. However, using the comparison of an $\Omega$-ring with a linear $\Omega$-algebra over the rational field, we obtain in § 3 a freeness theorem for $\Omega$-rings. The author expresses his indebtedness to A. G. Kurosh for valuable advice and remarks during the progress of the work, and for help in preparing the manuscript for the printer.