Irreducibility and Reduction Number of Ideals in a One Dimensional Analytically Irreducible Ring

Let R be a one dimensional analytically irreducible ring, and let I be an integral ideal of R. We study the irreducibility and the reduction number of I in relation with the corresponding semigroup ideal v(I) in v(R), where v(R) is the semigroup of values of R. It turns out that, if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We show also that the reduction number of I in R can assume any positive value less than the multiplicity of R and can be different from the reduction number of v(I).