Almost finitely generated inverse systems and reduced k-algebras

The purpose of this paper is to characterize one-dimensional local domains, or more in general reduced, in terms of its Macaulay's inverse system. This leads to study almost finitely generated modules in the divided power ring. We specialize the results to a numerical semigroup ring by computing explicitly its inverse system. In the graded case we characterize reduced arithmetically Gorenstein $0$-dimensional schemes.