On the Primality Property of Central Polynomials of Prime Varieties of Associative Algebras

In the paper, it is proved that, if $f(x_1,\dots,x_n)g(y_1,\dots,y_m)$ is a multilinear central polynomial for a verbally prime $T$-ideal $\Gamma$ over a field of arbitrary characteristic, then both polynomials $f(x_1,\dots,x_n)$ and $g(y_1,\dots,y_m)$ are central for $\Gamma$.