Gesetze in Ringen. I

Combinatorial Algebra: Syntax and Semantics

“…In 1950 Specht [306] suggested this problem for the special case of associative algebras over a field of characteristic 0 while the general case have been studied since the 1960s. Kemer [173], see also [174], proved that every variety of associative algebras over a field of characteristic 0 is finitely based thus solving Specht's problem.…”

Generic 2 × 2 matrices in positive characteristic

“…(We assume that 1 is a product of an empty set of commutators. The importance of the proper multilinear polynomials was discovered by Specht [20] who proved that over a field of characteristic 0 the polynomial identities of the unitary algebra R are equivalent to a set of multilinear proper identities. We use a stronger version of this result which states that the proper identities determine quite explicitly the structure and all numerical invariants of the relatively free algebras.…”

Kemer's theory for H-module algebras with application to the PI exponent

“…Two of the main problems in the theory of asssociative algebras satisfying a polynomial identity (PI in short) are the Specht problem (see [13]) and the Representability theorem ( [11]). The classical Specht problem asks whether a T -ideal can be generated as a T -ideal by a finite number of polynomials.…”