  We study prime monomial algebras. Our main result is that a prime finitely
presented monomial algebra is either primitive or it has GK dimension one and
satisfies a polynomial identity. More generally, we show this result holds for
the class of \emph{automaton algebras}; that is, monomial algebras that have a
basis consisting of the set of words recognized by some finite state automaton.
This proves a special case of a conjecture of the first author and Agata
Smoktunowicz.
