Absolutely Pure Modules

• Main Author: Pinzon, Katherine R.
• Format: Others
• Published: UKnowledge 2005
• Subjects: absolutely pure|modules|covers|localizations|derived functors
• Online Access: http://uknowledge.uky.edu/gradschool_diss/379; http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1382&context=gradschool_diss

Absolutely pure modules act in ways similar to injective modules. Therefore, through-out this document we investigate many of these properties of absolutely pure modules which are modelled after those similar properties of injective modules. The results we develop can be broken into three categories: localizations, covers and derived functors. We form S1M, an S1R module, for any Rmodule M. We state and prove some known results about localizations. Using these known techniques and properties of localizations, we arrive at conditions on the ring R which make an absolutely pure S1Rmodule into an absolutely pure Rmodule. We then show that under certain conditions, if A is an absolutely pure Rmodule, then S1A will be an absolutely pure S1Rmodule. Also, we dene conditions on the ring R which guarantee that the class of absolutely pure modules will be covering. These include R being left coherent, which we show implies a number of other necessary properties. We also develop derived functors similar to Extn R (whose development uses injective modules). We call these functors Axtn R, prove they are well dened, and develop many of their properties. Then we dene natural maps between Axtn(M;N) and Extn(M;N) and discuss what conditions on M and N guarantee that these maps are isomorphisms.