In the development of ring theory there are two strands, each with its own problems and methods, and although with many points of contact, they have never merged completely. One of them is the theory of algebras; this was a non-commutative (indeed sometimes even non-associative) theory from the beginning but there were always heavy finiteness restrictions, which are only gradually being relaxed. Thus while there is a fairly substantial theory of Artinian rings, the theory of Noetherian rings is still in its early stages, and although it is being developed vigorously, it is clear that some type of maximum condition is essential for its development.
A second and quite distinct line originated with the study of arithmetic in algebraic number fields. In the hands of (Cummer, Dedekind and E. Noether this led to the abstract notion of a Dedekind ring. Meanwhile, algebraic geometers found the need for affine rings hi the study of algebraic varieties; here Dedekind rings appear again, but as a rather special case (essentially the l-dimensional case). Now in the last few years our way of describing the geometrical notions has changed quite radically, with the result that the correspondence (rings)—>(varieties) has been extended and made precise: There is a contravariant functor from the category of commutative rings to the category of affine schemes, which is an equivalence. The effect of this connexion has been profound in both directions; we are concerned here particularly with the help afforded by the geometrical notions in studying rings...