  Irreducible decompositions of monomial ideals in polynomial rings over a
field are well-understood. In this paper, we investigate decompositions in the
set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A
is an arbitrary commutative ring with identity. We classify the irreducible
elements of this set, which we call m-irreducible, and we classify the elements
that admit decompositions into finite intersections of m-irreducible ideals.
