Vol 58, No 1 (2019)

Universal enveloping Lie Rota–Baxter algebras of pre-Lie and post-Lie algebras are constructed. It is proved that the pairs of varieties (RBLie, preLie) and (RBλLie, postLie) are PBW-pairs and that the variety of Lie Rota–Baxter algebras is not a Schreier variety.

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent. And its precise value is computed.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Whenever a lattice isomorphism φ implies an isomorphism between R and R^φ, we say that the ring R is determined by its subring lattice. The present paper is a continuation of previous research on lattice isomorphisms of finite rings. We give a complete description of projective images of prime and semiprime finite rings. One of the basic results is the theorem on lattice definability of a matrix ring over an arbitrary Galois ring. Projective images of finite rings decomposable into direct sums of matrix rings over Galois rings of different types are described.

We classify simple right-alternative unital superalgebras over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2. It is proved that such a superalgebra either is a Wall double W_2|2(ω), or is a Shestakov superalgebra S_4|2(σ) (characteristic 3), or is isomorphic to an asymmetric double, an 8-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double M₂[√1], an alternative 6-dimensional Shestakov superalgebra B_4|2 (characteristic 3), or an 8-dimensional Silva–Murakami–Shestakov superalgebra.