Let R be a Noetherian ring and M a finitely generated R-module. In this paper we introduce the set of prime ideals Fnd(M), the foundation primes of M. Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec(M), the annihilator primes of M, via Fnd(M). We show:(1) Fnd(M) is a finite set containing Annspec(M). Further, suppose that moreover every ideal of R has a centralizing sequence of generators ; now, Annspec(M) is equal to the set Ass(M) of associated primes of M. Then:(2) For an arbitrary P in Fnd(M), P is in Annspec(M) if and only if there is no Q in Annspec(M) such that P contains Q, and at the same time the minimal foundatin level on which appears P is greater than the minimal foundation level on which appears Q.