Hilbert rings with maximal ideals of different heights and unruly Hilbert rings

Abstract

Let f : R → S be a ring homomorphism and J be an ideal of S. Then the subring R f J := {(r, f (r) + j) | r ∈ R and j ∈ J} of R × S is called the amalgamation of R with S along J with respect to f . In this paper, we characterize when R f J is a Hilbert ring. As an application, we provide an example of Hilbert ring with maximal ideals of different heights. We also construct non-Noetherian Hilbert rings whose maximal ideals are all finitely generated (unruly Hilbert rings).