Rings of Endomorphisms

This dissertation reviews some results about rings of endomorphisms of modules, mainly In the form "if a module has the property IP then its ring of endomorphisms has the property Q". After an introductory Chapter 0, Chapter 1 is devoted to develop some concepts that will be necessary later on; a detailed study of the uniform (Goldie) dimension of a module is carried out and, in this vein, some original results of the author, which will appear elsewhere, are included in Section 4. In Chapter 2 we present the endomorphism ring of a module as well as a general technique for its study (Sections 5 and 6). The modules whose rings of endomorphisms have been reviewed are detailed next. In Section 7, injective and quasi-injective modules are considered; it is shown that the factor ring of their endomorphism ring modulo its radical is a regular and (right) self-injective ring. In Section 8, projective modules are discused; the Morita Theorem is recollected and some properties of a ring which are inherited by the endomorphism rings of its finitely generated projective modules are stated; also, a study of the projective modules with local endomorphism rings is done. In Section 9, we consider finite dimensional modules. First they are assumed to be also injective and, after dropping this hypothesis, we study the nilpotency of the nil subrings of their rings of endomorphisms; we also answer some questions about the quotient ring of the endomorphism ring of a finite dimensional nonsingular module. Finally, in Section 10, we look at what happens when the module is assumed to satisfy some chain conditions, in general at a first stage and under the hypothesis of quasi-injectivity or quasi-projectivity in the final paragraph of the dissertation