The unitary closure property of the prime varieties of associative algebras

We prove that every prime variety of associative algebras over an infinite field of characteristic $p>$0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity $x^{p^N}=0$ for sufficiently large $N$. We then describe all prime varieties in an interesting class of varieties of associative algebras.