This short survey article reviews our current state of understanding of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problems are listed. To the memory of my teacher and friend Brian Hartley

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...

A number of examples and constructions of local Noetherian domains without finite normalization have been exhibited over the last seventy-five years. We discuss some of these examples, as well as the theory behind them.

A number of results are proved concerning the Quillen ^-theory K+(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of 5. Applications are given to the computation of AT-groups of group algebras and of equivariant /^-theory for affine varieties.

for each  and i ≥ 0. The polynomial ring of integer-valued in rational  is defined by Int (    an important example binomial and is non-Noetherian ring. In this paper the algebraic structure rings has been studied their properties ideals. notion ideal generated a given set defined.  Which allows us to define new class Noetherian using ideals, which we named it binomiall...

Well-founded orders are the opposite of noetherian orders: every nonempty subset contains at least one minimal element. And a set is well-ordered when it is totally ordered by a wellfounded order: every nonempty set contains exactly one minimal element. Sofar, noetherian induction has been the most powerful way of proving properties inductively; it is indeed the most general one in the precise ...

We survey and extend recent work on integrally closed overrings of two-dimensional Noetherian domains, where such overrings are viewed as intersections of valuation overrings. Of particular interest are the cases where the domain can be represented uniquely by an irredundant intersection of valuation rings, and when the valuation rings can be chosen from a Noetherian subspace of the Zariski-Rie...