## 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 Rn

Rn = [[E, T], T] Qn ([T, [T, F]] [[E, T], T]) q-1 [T, [T, F]]
is infinitely based.
([. , .] denotes commutator, Q(x, y) = xp-1 yp-1 [x, y], p is the characteristic of the main field, q = pk. Qn = Q(x1, y1) ... Q(xn, yn).)

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, gi be a finite set of identities. The question is: is f a consequence of {gi } ?
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