Gomes, Catarina Araújo de Santa Clara
(1998)
Some generalizations of injectivity.
PhD thesis, University of Glasgow.
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Abstract
Chapter 1 covers the background necessary for what follows. In particular, general properties of injectivity and some of its wellknown generalizations are stated.
Chapter 2 is concerned with two generalizations of injectivity, namely near and essential injectivity. These concepts, together with the notion of the exchange property, prove to be a key tool in obtaining characterizations of when the direct sum of extending modules is extending.
We find sufficient conditions for a direct sum of two extending modules to be extending, generalizing several known results. We characterize when the direct sum of an extending module and an injective module is extending and when the direct sum of an extending module with the finite exchange property and a semisimple module is extending. We also characterize when the direct sum of a uniformextending module and a semisimple module is uniformextending and, in consequence, we prove that, for a right Noetherian ring R, an extending right Rmodule M1 and a semisimple right Rmodule M2, the right Rmodule M1 M2 is extending if and only if M2 is M1/Soc(M1)injective.
Chapter 3 deals with the class of selfcinjective modules, that can be characterised by the lifting of homomorphisms from closed submodules to the module itself.
We prove general properties of selfcinjective modules and find sufficient conditions for a direct sum of two selfcinjective to be selfcinjective. We also look at selfcuinjective modules, i.e. modules M such that every homomorphism from a closed uniform submodule to M can be lifted to M itself.
We prove that every selfcinjective free module over a commutative domain that is not a field is finitely generated and then proceed to consider torsionfree modules over commutative domains, as was done for extending modules in [31].
We also characterize when, over a principal ideal domain, the direct sum of a torsionfree injective module and a cyclic torsion module is selfcuinjective.
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