Stefano Marseglia: Super-multiplicativity of ideal norms in number fields

Let R be a commutative ring, I an R-ideal. The norm of I is defined as N(I) = #(R/I) = [R : I]. We say that the norm is super-multiplicative on R (briefly R is S.M.) if for every pair of ideals I; J s.t. [R : IJ] < 1 we have N(IJ) \ge N(I)N(J). Observe that if the norm of an order R satisfies the other inequality for every maximal ideal p, that is N(p^2) \le N(p)^2, then dim_(R/p)(p/p^2) = 1, i.e. the order is Dedekind and the ideal norm is actually multiplicative. The main goal of my thesis is to give a characterization for a number ring, that is a subfield of a number field, of being S.M. in terms of the minimal number of generators of its ideals. I proved: Theorem. Let R be a number ring. Then the following statements are equivalent: (1) every ideal of R can be generated by 3 elements; (2) for every ring extension R \subseteq R'\subseteq R~, where R~ is the normalization of R, we have that the norm is super-multiplicative.