No abstract

Irreducibility of ideals in a one-dimensional analytically irreducible ringAbstract Let R be a one-dimensional analytically irreducible ring and let I be an integral ideal of R. We study the relation between the irreducibility of the ideal I in R and the irreducibility of the corresponding semigroup ideal v(I). It turns out that if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal.A numerical semigroup is a subsemigroup of N, with zero and with finite complement in N. The numerical semigroup generated bywill denote the maximal ideal of S, e the multiplicity of S, that is the smallest positive integer of S, g the Frobenius number of S, that is the greatest integer which does not belong to S. A relative ideal of S is a nonempty subset I of Z such that I + S ⊆ I and I + s ⊆ S, for some s ∈ S. A relative ideal which is contained in S is an integral ideal of S. If I, J are relative ideals of S, then the following are relative ideals too:An integral ideal I of a numerical semigroup S is called irreducible if it is not the intersection of two integral ideals which properly contain I. Consider the partial order on S given by s 1 s 2 ⇔ s 1 + s 3 = s 2 , for some s 3 ∈ S, and for s ∈ S, set B(s) = {x ∈ S|x s}.

No abstract