  In general, representation rings are well-known as Green rings from module
categories of Hopf algebras.
  In this paper, we study Green rings in the context of monoidal categories
such that representations of Hopf algebras can be investigated through Green
rings of various levels from module categories to derived categories in the
unified view-point. Firstly, as analogue of representation rings of Hopf
algebras, we set up the so-called Green rings of monoidal categories, and then
list some such categories including module categories, complex categories,
homotopy categories, derived categories and (derived) shift categories, etc.
and the relationship among their corresponding Green rings.
  The main part of this paper is to characterize representation rings and
derived rings of a class of inite dimensional Hopf algebras constructed from
the Nakayama truncated algebras $KZ_{n}/J^{d}$ with certain constraints. For
the representation ring $r(KZ_{n}/J^{d})$, we completely determine its
generators and the relations of generators via the method of Pascal triangle.
For the derived ring $dr(KZ_{n}/J^{2})$(i.e., $d=2$), we determine its
generators and give the relations of generators. In these two aspects, the
polynomial characterizations of the representation ring and the derived ring
are both given.
