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

Let f : R -> S be a ring homomorphism and J be an ideal of S. Then the subring R (sic)(f) J := {(r, f (r) + j) vertical bar r is an element of R and j is an element of J} of R x S is called the amalgamation of R with S along J with respect to f. In this paper, we characterize when R (sic)(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).