Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of Mn(F) was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for Mn(F) and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider Mn(R), where R admits a regular grading, with a grading such that Mn(F) is a homogeneous subalgebra and provide sufficient conditions - satisfied by Mn(E) with the trivial grading - to prove that Mn(R) has the primeness property if Mn(F) does. We also prove that the algebras Ma,b(E) satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.