Some extremal varieties of associative algebras
by
Elena Kireeva
Moscow Pedagogical State University, Russia.
Coauthors: Alexei Krasilnikov

Let F be a field, A be a free associative algebra over F on free generators x₁, x₂, ... An element v=v(x₁, ... , x_n) of the algebra A is called a polynomial identity or identity in an associative F-algebra G if v(g₁, ... , g_n)=0 for all elements g₁, ... , g_n of the algebra G. In this case an expression v=0 is also called an identity. The class of all associative F-algebras satisfying a given set of identities is called a variety. An ideal V in the algebra A is called T-ideal if V is a fully invariant ideal, that is V is closed under all endomorphisms of A. It is known that there is one-to-one correspondence between the set of all varieties of associative F-algebras and the set of all T-ideals in A. Namely if V is a variety of associative F-algebras then the corresponding T-ideal is the ideal of all identities which are satisfied in every algebra of V and if V is a T-ideal in A then the corresponding variety is the variety of all F-algebras satisfying every identity of V. The quotient algebra A/V is called a relatively free algebra (of countable infinite rank) of the variety V. We refer to [1], [4], [5] and [8] for the terminology and basic facts concerning varieties of associative algebras and polynomial identities.

Let V be a T-ideal in A. A vector subspace U in A/V is called a T-space if U is a fully invariant subspace, that is U is closed under all endomorphisms of the algebra A/V. A T-space U is finitely generated if there exist a finite set of elements u₁, ... , u_k of U such that every element u of U can be written as a linear combination of the elements f₁(u₁), ... , f_k(u_k) for some endomorphisms f₁, ... , f_k of the algebra A/V.

Let F be a field of characteristic p > 0 and let V_p be the variety of associative F-algebras given by the identities [[x, y], z]=0 and x^p=0 if p > 2, and by the identities [[x, y], z]=0 and x⁴=0 if p=2 (where [x, y]=xy-yx). It was proved by A.V. Grishin [6] for p=2 and by V.V. Shchigolev [10] for p > 2 that the relatively free algebra of countable infinite rank of the variety V_p contains non-finitely generated T-spaces. The construction of such T-spaces is one of the most important steps in the solution of the Specht problem over a field of a prime characteristic (see [2], [6], [9] and also [3], [7], [10]).

Our main result is as follows.

Theorem Let F be a field of characteristic p > 0. Then V_p is a minimal variety of associative F-algebras whose relatively free algebras of countable infinite rank contain non-finitely generated T-spaces.

References

[1] Yu.A. Bakhturin and A.Yu. Olshanskii, Identities, Algebra II, Encyclopedia of Mathematical Sciences 18, Springer, Berlin, 1991, 107-221.

[2] A.Ya. Belov, On non-Specht varieties, Fundam. Prikl. Mat. 5 (1999), 47-66 (Russian).

[3] A.Ya. Belov, Counterexamples to the Specht problem, Mat.Sb. 191 (2000), 13-24 (Russian). Sb. Math. 191 (2000), 329-340.

[4] L.A. Bokhut, I.V. Lvov and V.K. Kravchenko, Noncommutative rings, Algebra II, Encyclopedia of Mathematical Sciences 18, Springer, Berlin, 1991, 1-106.

[5] V. Drensky, Free algebras and PI-algebras, Springer, Singapore, 2000.

[6] A.V. Grishin, Examples of T-spaces and T-ideals of characteristic 2 without the finite basis property, Fundam. Prikl. Mat. 5 (1999), 101-118 (Russian).

[7] A.V. Grishin, On non-Spechtianness of the variety of associative rings that satisfy the identity x³²=0, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 50-51.

[8] L.H. Rowen, Ring Theory, Academic Press, Boston, 1991.

[9] V.V. Shchigolev, Examples of infinitely based T-ideals, Fundam. Prikl. Mat. 5 (1999), 307-312 (Russian).

[10] V.V Shchigolev, Examples of infinitely basable T-spaces, Mat.Sb. 191 (2000), 143-160 (Russian). Sb. Math. 191 (2000), 459-476.

Partially supported by INTAS.

Date received: April 30, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cali-03.