  In this article we study inequalities of ideal norms. We prove that in a
subring $R$ of a number field every ideal can be generated by at most $3$
elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for
every pair of non-zero ideals $I$ and $J$ of every ring extension of $R$
contained in the normalization of $R$.
