On varieties of quasiregular algebras.

*(Russian)*Zbl 0578.16006An 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)].

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 |