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  • Print publication year: 2009
  • Online publication date: September 2011

11 - Characterizing endomorphisms

Summary

We fix for the duration of this chapter a ring R and a right R-module G.

The theorems in Chapter 2 raise a pair of interesting questions. (1) Which properties of G can be described in terms of EndR(G). (2) Which properties of EndR(G) can be described in terms of G?

In this chapter we will use the equivalences in Chapter 2 to provide partial answers to these questions. Specifically, we will consider the homological dimensions of G as a left EndR(G)-module, the right global dimension of EndR(G), rgd(EndR(G)), the flat left dimension fd(G), and of the injective dimension idE(G) of the left EndR(G)-module G. Satisfying results are given for semi-simple Artinian endomorphism rings (rgd(EndR(G)) = 0), right hereditary endomorphism rings (rgd(EndR(G)) ≤ 1), for right Noetherian right hereditary endomorphism rings, and for rings EndR(G) such that rgd(EndR(G)) ≤ 3. The answers are most complete when G is a self-small or self-slender right R-module.

Furthermore, there are quite a few new terms used in this chapter. To help the reader organize these terms, we have included a small Glossary at the end of the chapter, and some exercises detailing implications among these new terms.

Flat endomorphism modules

Theorem 10.18 is an illuminating result. It explains among other things the right ideals of EndR(G) in terms of the G-generated R-submodules of the self-small faithfully E-flat right R-module G.