Nonaxiomatizability of lattice-orderable rings

Two elementarily equivalent rings, one of which is lattice-orderable and the other is not lattice-orderable, are constructed. Hence follows the elementary non closedness and the nonaxiomatizability of the class of all lattice-orderable rings. This example shows that the class of all lattice-orderable rings is nonaxiomatizable in the class of directedly orderable rings.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 449–452, April, 1977.     

Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly $\pi$-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element $a\in R$ is called pseudopolar if there exists $p\in R$ such that $p^2=p\in {\mathrm{comm}}^2(a)\text{,}a+p\in U(R)\mathrm{and}{a}^{k}p\in J(R)$ for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly $\pi$-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.     

Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type

We construct the Grothendieck rings of a class of 2n²dimensional semisimple Hopf Algebras H_2n²,which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H_2n²-modules are classified.Furthermore,we describe the Grothendieck rings r(H_2n²)by generators and relations explicitly.     

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

Transvection parameter sets in linear groups over associative division rings and special Jordan algebras defined by these rings

In this paper we construct a correspondence between a class of irreducible linear groups of finite degree over an associative division ring D and special Jordan rings which are defined by D. Received: 14 September 2005     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Permutation orbifolds and associative algebras

Let V be a vertex operator algebra and g =(1 2 ··· k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V?k. It is proved that Dong-Li-Mason’s associated associative algebra Ag(V?k) is isomorphic to Zhu’s algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V?k-modules and irreducible V-modules.     

Power associative composition algebras

Composition algebras of arbitrary dimension over a field and satisfying the identities x ² x=x x ² and (x ²)²=(x ² x)x are shown to be precisely the well-known unital composition algebras, with the exception of three two dimensional algebras over the field of two elements. Received: 1 February 2000     

On rings asymptotically close to associative rings

The subject of this work is an extension of A. R. Kemer’s results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings). We prove the finite-basedness of systems of identities (the Specht property), the representability of finitely-generated relatively free algebras, and the rationality of their Hilbert series. For this purpose, we extend the Razymslov-Zubrilin theory to Kemer polynomials. For a rather broad class of varieties, we prove Shirshov’s theorem on height.