Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings

Abstract

Much of Chapter 4 is devoted to the exposition of the structure theory of simple right Noetherian rings. The basic tools generalize the theorems of Morita [58] characterizing similarity of two rings A and B, that is, when there is an equivalence mod-A ≈ mod-B of categories. Morita’s characterization 4.29 predicates the existence of a finitely generated projective module P which is a generator of the category mod-B such that A is isomorphic to End B P. In the Morita situation 4.30 there is a lattice isomorphism $$\left\{ {\begin{array}{*{20}{c}} {right A - submodules of \to right ideals of B} {IP \leftrightarrow I} \end{array}} \right.$$ sending (B, A)-submodules onto ideals of B. More generally, 4.7, if U is any finitely generated projective faithful left B-module over any ring B, and A = End B U then there is a lattice isomorphism $$\left\{ {\begin{array}{*{20}{c}}{right A - submodules of U \to (right ideas of V) T} {IU \leftrightarrow I = IT} \end{array}} \right.$$ where T is the trace of U in B (Correspondence Theorem for Projective Modules.) This isomorphism sends (B, A)-submodules of U into (ideals of B) T. Thus, when T = B (and only then) the two theorems coincide.