Let d 1 , … , d r be pairwise relatively prime positive square-free integers, with d j ≥ 2 for all 1 ≤ j ≤ r . By using the elementary matrix methods, we give a complete classification of the irreducible Z + -modules over the domain Z [ d 1 , … , d r ] . Moreover, all of these irreducible Z + -modules are explicitly constructed.