No associative $PI$-algebra coincides with its commutant

We prove that each (possibly not finitely generated) associative $PI$-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every $T$-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of $T$-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of $T$-prime varieties).