Partially ordered sets and nonmatrix identities of associative algebras
In this paper the following theorem including almost all the previous results on Specht's problem is proved. Theorem. Every \(T\)-ideal of the free associative algebra \(F \langle x_1, x_2, \dots \rangle\) over a field \(F\) of characteristic zero containing the polynomial \([x_1,x_2]\dots [x_{2n-1}, x_{2n}]\) finitely generated as \(T\)-ideal. In other words, the variety of all associative algebras satisfying the identity \([x_1,x_2]\dots [x_{2n-1}, x_{2n}] = 0\) satisfies the minimum condition for subvarieties. Corollary 1. Let \(A\) be a finitely generated \(F\)-algebra satisfying the identity which is false on the matrix algebra \(F_2\). Then \(A\) has a finite basis of identities. Corollary 2. The variety generated by the algebra \(F_2\) satisfies the minimum condition for subvarieties.
scientific article; zbMATH DE number 3557987