  Let $R$ be the power series ring or the polynomial ring over a field $k$ and
let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein
$k$-algebras $R/I$ are in one-to-one correspondence with the cyclic
$R$-submodules of the divided power series ring $\Gamma. $ The result is
effective in the sense that any polynomial of degree $s$ produces an Artinian
Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors
extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma
$ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras.
However, these submodules in positive dimension are not finitely generated. Our
goal is to give constructive and finite procedures for the construction of
Gorenstein $k$-algebras of dimension one and any codimension. This has been
achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $
Applications to the Gorenstein linkage of zero-dimensional schemes and to
Gorenstein affine semigroup rings are discussed.
