Noetherian rings of non-local rank

Abstract: The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$, among all ideals $I$ in $R$. In this paper, we obtain a characterization of Noetherian rings $R$ whose rank is not equal to the supremum of ranks of localizations of $R$ at maximal ideals. It turns out that any such ring is a direct product of a finite number of local principal Artinian rings and Dedekind domains, at least one of which is not principal ideal ring. As an application, we show that the rank of the ring of polynomials over an Artinian ring can be computed locally.