On identities and varieties of associative algebras of finite type. (English. Russian original) Zbl 1144.16309

Vasil’ev, S. K.

J. Math. Sci., New York 130, No. 3, 4651-4664 (2005); translation from Zap. Nauchn. Semin. POMI 305, 18-43 (2003).

Summary: Varieties in which each finitely generated relatively free algebra is finitely presented are considered. A description of such varieties with identities is obtained, and a number of their structural properties are found.

MSC:

16R10	\(T\)-ideals, identities, varieties of associative rings and algebras
16S10	Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
08B20	Free algebras

Keywords:

varieties of finite type; finitely generated relatively free algebras; finitely presented algebras; ideals of identities

References:

[1]	V. T. Markov, ”On generating sets of T-ideals of finitely generated free algebras,” Algebra Logica, 18, No.5, 587–598 (1979).
[2]	S. I. Kublanovsky, ”On varieties of associative algebras with local conditions of finiteness,” Algebra Logica, 9, No.4, 119–174 (1997).
[3]	O. G. Kharlampovich and M. V. Sapir, ”Algorithmic problems in varieties,” Internat. J. Algebra Comput., 5, Nos.4–5, 379–602 (1995); Algebra Logica, 9, No. 4, 119–174 (1997). · Zbl 0837.08002
[4]	I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs (1968). · Zbl 0177.05801
[5]	N. Jacobson, Structure of Rings, AMS (1956). · Zbl 0073.02002
[6]	S. I. Kublanovsky, ”Locally finitely approximate and locally representable varieties of associative rings and algebras,” Leningrad State Pedagogical Institute, Leningrad (1982) (deposited at VINITI, No. 6143-82).
[7]	N. Bourbaki, Algebre, Chaps. IV and V.