Factorization in K[[S]]

Let K be a field, S = 〈n ₁,…, n _r 〉 a numerical semigroup, and K[[S]] the (power series) semigroup ring. In this paper, we study lengths of factorizations in K[[S]]. We show that n _r /n ₁ ≤ ρ(K[[S]]) ≤ (g(S) + n ₁)/n ₁, where g(S) is the Frobenius number of S. This inequality is an equality when g(S) + n ₁ = n _r , but in general ρ(K[[S]]) may lie strictly between the two bounds.