Note on Shimoda rings and Shimoda’s conjecture

A Noetherian local ring (R, M, k) is called a Shimoda ring if every prime ideal in the punctured spectrum is of the principal class; that is, the minimal number of generators \(\mu (P)\) $\mu (P)$ of every nonmaximal prime ideal P of R is equal to the height ht(P) of P. This paper extends the notion of Shimoda ring to commutative rings that are not Noetherian and seeks for classes of domains where Shimoda’s conjecture is valid.