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Algebra pp 185-229 | Cite as

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

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

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.

Keywords

Left Ideal Projective Module Regular Element Endomorphism Ring Quotient Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

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