Specht type problems and related questions.
by
Alexei Kanel-Belov
Moscow Center of Continious Mathematical Education

There was a problem, posed by W.Specht in 1950:

Specht problem Does any increasing chain of T-ideals stabilise? Or - Is any set of identities finitely based (can be presented by a finite subset)?

W.Specht had in mind the case of algebras over a field of zero characteristic. This problem was solved affirmatively by A.R.Kemer. A.I.Maltzev gave another point of view. He considered the more general case (any characteristic and over an arbitrary ring).

The author proved the next theorem:

Theorem 1 The following set of identities R_n

Rn=[[E, T], T]Qn([T, [T, F]][[E, T], T])q-1[T, [T, F]]

is infinitely based. ([·, ·] denotes commutator, Q(x, y)=x^p-1y^p-1[x, y], p is characteristic of main field, q=p^k. Q_n=Q(x₁, y₁) ... Q(x_n, y_n).

The finite basis problem can be considered in the local case (i.e. chain conditions on sets of identities in a finitely generated algebra). There was a well known problem, related to this question:

Does any increasing chain of T-ideals in a finitely generated algebra stabilize?

Is any finitely generated relatively free algebra representable? (i.e. embedable in a matrix algebra over a Notherian commutative ring)

Can any relatively free PI-algebra be approximated by finite dimentional ones? These problems where posed by L.Bokut', I.Lvov, A.I.Maltzev. A.R.Kemer obtained a positive answer in the homogeneous case, i.e. when the main field is infinite.

Theorem 2 (Belov) Every relatively free PI-ring is representable, and any increasing chain of ideals of identities in a finitely generated ring stabilizes.

There was a problem, posed by A.I.Maltzev in 1967 (and also by P.Cohn, Tarski):

Maltzev problem Let f be an identity, g_i be a finite set of identities. The question is: is f a consequence of {g_i}? Does a general algorithm solving this question exist?

In case of groups the answer is ``No'' (It was shown by Yu.Kleiman). The author proved the next result.

Theorem 4 There exists such general algorithm in the case of associative rings.

The Specht-type problems are closely connected with properties of Hilbert series of algebras. C.Procesi posed a problem about rationality of Hilbert series of algebra of general matrices. In case of 2×2 matrices this problem was solved by V.Drensky. He also proved rationality of Hilbert series of relatively free algebras in non-matrix varieties.

Theorem 5 a) The Hilbert series of relatively free algebras is rational.

b) There is a representable algebra with a transcendental Hilbert series.

Acknowlegments The author is grateful to Prof. A.V.Michalev and all the members of the seminar ``Ring Theory'' in Moscow State University for moral support and great interest. The author is specially grateful to Prof. V.N.Latyshev and A.R.Kemer for useful discussions.

References

1. A.R.Kemer. Ideals of identities of associative algebras. Dr.Sci.Thesis.

2. A.R.Kemer. The identities of finitely generated PI-algebras over an infinite field. Izv.AN.USSR., 1990, 54, N4, pp.726-753.

3. A.J.Belov, V.V.Borisenko, V.N.Latyshev. Monomial algebras. NY, Plenum, 1998.

4. A.J.Belov. About rationality of Hilbert series of relatively free algebras. UMN, 1997, v.52, n4.pp.153-154.

5. A.J.Belov. About non-Specht varieties. Fund i prikl.matem., 1999, n5, v1, pp.47-66.

Date received: December 29, 2001