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**Ideals of identities of associative algebras. Transl. from the Russian by C. W. Kohls. Transl. ed. by Ben Silver.**
*(English)*
Zbl 0732.16001

Translations of Mathematical Monographs, 87. Providence, RI: American Mathematical Society (AMS). v, 81 p. $ 75.00 (1991).

The present monograph is devoted to the solution of one of the most famous questions in algebra namely Specht’s problem. In 1950 W. Specht [Math. Z. 52, 557-589 (1950; Zbl 0032.38901)] asked whether every associative algebra over a field of characteristic zero has a finite basis of identities, i.e. whether every variety of associative algebras is finitely based. Analogous problems for other algebraic systems were solved as follows. In 1970 A. Yu. Ol’shanskij [Izv. Akad. Nauk SSSR, Ser. Mat. 34, 376-384 (1970; Zbl 0215.105)] gave a negative answer for varieties of groups. M. R. Vaughan-Lee [Q. J. Math., Oxf. II. Ser. 21, 297-308 (1970; Zbl 0204.35901)] and V. S. Drensky [Algebra Logika 13, 265-290 (1974; Zbl 0298.17011)] solved negatively the problem for Lie algebras over a field of finite characteristic. On the other hand, I. V. L’vov [Algebra Logika 12, 667-688 (1973; Zbl 0288.16009)] and R. L. Kruse [J. Algebra 26, 298-318 (1973; Zbl 0276.16014)] showed that each finite ring has a finite basis of identities. A. Ya. Vajs and E. I. Zel’manov [Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No.6, 42-51 (1989; Zbl 0679.17013)] proved that every finitely generated Jordan algebra over a field of characteristic zero is Spechtian. A. V. Il’tyakov [On varieties of representations of Lie algebras (Akad. Nauk SSSR, Siberian Division, Inst. Math., Novosibirsk, Preprint No.9) (1991) and Specht’s property of varieties of PI-representations of finitely generated Lie algebras over a field of characteristic zero (ibid. Preprint No.10) (1991)] solved the problem in the case of finite-dimensional Lie algebras (and in a more general situation). Note that the last two results on Jordan and Lie algebras were obtained using the ideas and methods of the present monograph.

Many algebraists dealt with similar problems for some concrete varieties of associative algebras, see e.g. the references and the summaries of the surveys of Yu. A. Bakhturin and A. Yu. Ol’shanskij [Identities, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 18, 117-240 (1988; Zbl 0673.16001)] and of Yu. P. Razmyslov [Identities of algebras and their representations, Sovremennnaya Algebra, 14, Moscow, Nauka (1989; Zbl 0673.17001)].

In a series of papers the author of this monograph obtained a number of strong and significant results on varieties of associative algebras. In [Sib. Mat. Zh. 19, 54-69 (1978; Zbl 0385.16009)] he showed that every variety of associative algebras with exponential growth of codimensions is Spechtian. Then [Algebra Logika 19, 255-283 (1980; Zbl 0467.16025)] he described all non-matrix varieties. In his fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 48, 1042-1059 (1984; Zbl 0586.16010)] he investigated varieties of \({\mathbb{Z}}_ 2\)-graded algebras and prime varieties and showed that the solution of Specht’s problem would follow from that of some concrete \({\mathbb{Z}}_ 2\)-graded algebras. Finally [Algebra Logika 26, 597-641 (1987; Zbl 0646.16013)] he gave the positive solution of this problem using varieties of \({\mathbb{Z}}_ 2\)-graded algebras. In the paper [Algebra Logika 27, 274-294 (1988; Zbl 0678.16012)] he proved that the relatively free algebra of a non-Grassmann variety is representable by matrices.

All these results were obtained using the idea of \({\mathbb{Z}}_ 2\)-grading and superalgebras and the description of the prime T-ideals and varieties given by the author in the papers cited above. The present monograph is a translation of his dissertation and covers all these remarkable achievements. Note that the monograph is written in the best way and I would recommend it to all algebraists especially to those who have not read the papers cited above but are interested in the theory of PI- algebras. It is worth noticing that the papers of A. R. Kemer discussed above were (to a certain extend) written in a difficult and hard manner. On the contrary, the present monograph is arranged in such a way that makes reading easy and enjoyable. The idea of translating and publishing this monograph is welcome.

Many algebraists dealt with similar problems for some concrete varieties of associative algebras, see e.g. the references and the summaries of the surveys of Yu. A. Bakhturin and A. Yu. Ol’shanskij [Identities, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 18, 117-240 (1988; Zbl 0673.16001)] and of Yu. P. Razmyslov [Identities of algebras and their representations, Sovremennnaya Algebra, 14, Moscow, Nauka (1989; Zbl 0673.17001)].

In a series of papers the author of this monograph obtained a number of strong and significant results on varieties of associative algebras. In [Sib. Mat. Zh. 19, 54-69 (1978; Zbl 0385.16009)] he showed that every variety of associative algebras with exponential growth of codimensions is Spechtian. Then [Algebra Logika 19, 255-283 (1980; Zbl 0467.16025)] he described all non-matrix varieties. In his fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 48, 1042-1059 (1984; Zbl 0586.16010)] he investigated varieties of \({\mathbb{Z}}_ 2\)-graded algebras and prime varieties and showed that the solution of Specht’s problem would follow from that of some concrete \({\mathbb{Z}}_ 2\)-graded algebras. Finally [Algebra Logika 26, 597-641 (1987; Zbl 0646.16013)] he gave the positive solution of this problem using varieties of \({\mathbb{Z}}_ 2\)-graded algebras. In the paper [Algebra Logika 27, 274-294 (1988; Zbl 0678.16012)] he proved that the relatively free algebra of a non-Grassmann variety is representable by matrices.

All these results were obtained using the idea of \({\mathbb{Z}}_ 2\)-grading and superalgebras and the description of the prime T-ideals and varieties given by the author in the papers cited above. The present monograph is a translation of his dissertation and covers all these remarkable achievements. Note that the monograph is written in the best way and I would recommend it to all algebraists especially to those who have not read the papers cited above but are interested in the theory of PI- algebras. It is worth noticing that the papers of A. R. Kemer discussed above were (to a certain extend) written in a difficult and hard manner. On the contrary, the present monograph is arranged in such a way that makes reading easy and enjoyable. The idea of translating and publishing this monograph is welcome.

Reviewer: P.Koshlukov (Sofia)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16Rxx | Rings with polynomial identity |

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

16W50 | Graded rings and modules (associative rings and algebras) |

16W55 | “Super” (or “skew”) structure |