Let \({\mathfrak m}\) be an element of an Abelian monoid, and let \(\Omega({\mathfrak m})\) denote the total number of prime elements (counted with multiplicity) generating \({\mathfrak m}\) . We investigate the distribution of \(\Omega({\mathfrak m})\) over the subsets of h-free and h-full elements, obtaining moment estimates and establishing its normal order within these subsets. This extends the authors’ previous work (see S. Das et al., 2025c) on \(\omega({\mathfrak m})\) , where multiplicities of prime elements were not considered. In particular, we develop new identities involving sums over prime elements, which play a central role in the analysis. Several applications are presented, including ideals in number fields, effective divisors in global function fields, and effective zero-cycles on geometrically irreducible projective varieties.
