Аннотация:The following problem was posed by W.Specht in 1950:
Specht problem. Does any increasing chain of T-ideals stabilise?
Equivalently: Is any set of identities finitely based (can be
presented by a finite subset)?
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 speaker proved:
Theorem 1. The set of identities
{R_n=[[E,T],T]Q_n([T,[T,F]][[E,T],T])^{q-1}[T,[T,F]]}
is infinitely based.
([ , ] denotes commutator, Q(x,y)=x^{p-1}y^{p-1}[x,y],
p is the characteristic of the base field, q=p^k,
Q_n=Q(x_1,y_1) ... 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
were well known problems 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. embeddable in a matrix algebra over a Noetherian commutative ring)
Can any relatively free PI-algebra be approximated by finite
dimensional ones?
These problems where posed by L.Bokut', I.V.Lvov, and A.I.Maltzev.
A.R.Kemer obtained a positive answer in the homogeneous case, i.e.
when the base field is infinite.
The speaker proved these results for algebras over an arbitrary Noetherian
commutative ring.
Theorem 2. Every relatively free PI-ring is representable, and
any increasing chain of ideals of identities in a finitely generated
ring stabilises.
The following problem was posed by A.I.Maltzev in 1967 (and also by
P.Cohn, Tarski):
Maltzev problem. Let f be an identity, and let {g_i} be a finite set of
identities. Does there exist a general algorithm solving the question:
is f a consequence of {g_i}?
In case of groups the answer is ``No'' (This was shown by Yu.Kleiman).
Theorem 3. There exists such a 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 by 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 4. a) The Hilbert series of relatively free algebras are rational.
b) There is a representable algebra with a transcendental Hilbert series.
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.
