On varieties of quasiregular algebras. (Russian) Zbl 0578.16006

Andrunakievich, V. A.; Ryabukhin, Yu. M.

Mat. Sb., N. Ser. 127(169), No. 4(8), 419-444 (1985).

An associative algebra is quasiregular if it is a group with respect to the adjoint multiplication \(x\circ y=x+y-xy\). The class \({\mathfrak A}^*\) of all such algebras is a variety of algebraic systems. An important object related to the subvariety \({\mathfrak M}^*\subset {\mathfrak A}^*\) is the variety of associative algebras \({\mathfrak M}=var {\mathfrak M}^*\). For any algebra satisfying the adjoint Ore condition (with respect to the operation \(x\circ y)\), one defines an algebra of adjoint fractions. The adjoint Ore theorem holds. \({\mathfrak M}^*\) is called an Ore variety, if the relatively free algebra of \({\mathfrak M}\) satisfies the adjoint Ore condition.
In the paper under review the authors initiate a systematic study of the varieties of quasiregular algebras. The main purpose is to investigate Ore varieties. Necessary and sufficient conditions are given for a variety to have the Ore property. In particular, any such variety can be defined by an ordinary system of polynomial identities. The relatively free and the prime algebras are examined. It is proved that if the algebras from \({\mathfrak M}^*\) are nilpotent groups (with respect to the operation \(x\circ y)\), then \({\mathfrak M}^*\) is an Ore variety. Numerous consequences of the main results are obtained and many examples are given. A list of open problems is enclosed. For other results concerning the varieties of quasiregular algebras, see the authors [Algebra Logika 23, 605-623 (1984)].

Reviewer: V.Drensky

MSC:

16Rxx	Rings with polynomial identity
16N60	Prime and semiprime associative rings
16N40	Nil and nilpotent radicals, sets, ideals, associative rings
08B20	Free algebras

Keywords:

adjoint multiplication; relatively free algebra; Ore condition; varieties of quasiregular algebras; Ore varieties; polynomial identities; prime algebras