[For the entire collection see Zbl 0741.00019.]
This is a survey of what is known about the identities satisfied by associative algebras, including several major results of the author. Let \(R\) be an associative algebra over a field \(F\), let \(S\) be the free \(F\)- algebra on a countable set, and let \(T(R)\) be the ideal of \(S\) consisting of those elements of \(S\) which are identities for \(R\). Suppose further that \(R\) satisfies a polynomial identity, i.e. \(T(R)\neq 0\). The author showed that if \(R\) is finitely-generated and \(F\) is infinite, then \(T(R)=T(C)\) for some finite-dimensional \(F\)-algebra \(C\); thus \(R\) satisfies the same identities as some finite-dimensional algebra. Part of the proof of this result is sketched. A similar classification is given for the case in which \(R\) is not necessarily finitely-generated, provided that \(F\) has characteristic 0, and in this context the author also solved Specht’s problem by showing that there is a finite base for the identities satisfied by \(R\). Much less is known when \(F\) has characteristic which is not zero, but the author gives his proof that in this case \(R\) satisfies a standard polynomial identity, thus solving a problem posed by Volichenko.