Irreducibility of monomial ideals

Question

So far, I have encountered two notions of irreducibility for ideals:

1. We say that an ideal $I$ in a commutative ring $R$ is irreducible if it cannot be expressed as an intersection of two strictly bigger ideals, that is, if there are no ideals $J,K$ such that $I=J\cap K$, with $I\subsetneq J$ and $I\subsetneq K$.

2. When $I$ is a monomial ideal in a polynomial ring, we say that $I$ is irreducible if it can't be expressed as an intersection of two strictly bigger monomial ideals.

Are thes two definitions equivalent for a monomial ideal $I$? Clearly 1 implies 2, but I'm not sure about the converse.