**Specht type problems and related questions**

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)?

(An identity is a polynomial, which is identically zero on an algebra;
the set of
identities is an ideal in the ring of non-commutative polynomials which
is stable under substitutions. The set of identities is finitely based if
it is generated by a finite subset under substitutions, linear
combinations and multiplications by arbitrary elements.)

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 in 1967 gave another point of view.
He considered the most general case (any characteristic
and over an arbitrary ring).

The author proved the next theorem:

**Theorem.**

The following set of identities *R*_{n}

*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 main 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 some other open questions related with the Specht problem.

- Does any increasing chain of ideals of identities 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.Lvov, A.I.Maltzev.
A.R.Kemer obtained a positive answer in the homogeneous case, i.e.
when the main field is infinite.

The author solved these questions for algebras over arbitrary noetherian
commutative rings.

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

Let *f* be an identity, *g*_{i} be a finite set of identities.
The question is: is *f* a consequence of {*g*_{i }} ?

**Maltzev problem.**

Does a general algorithm solving this question exist?

In case of groups the answer for a similar question is ``No''
(as was shown by Yu.Kleiman).
The author proved the next result.

**Theorem.**

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

The Specht type problems give a new point of view on non-commutative
algebraic geometry which will be discussed.

*A. J. Belov*