On The Theory Of Abstract R-vector Spaces Over A Commutative Regular Ring

The notion of an Abstract Boolean Vector space (an Abstract BVectorrnspace) is introduced by Subrahmanyam N.V. and he studiedrnintensively on this spaces. Later N.Raja Gopala Rao introducedrnthe concept of an Abstract R-Vector spaces as a generalization ofrnAbstract Boolean Vector space of Subrahmanyam N.V. He introducedrnthe notion of linear endomorphisms and a_ne transformationsrnin Abstract R-Vector spaces and studied its properties. Further,rnhe made a study on the geometric aspect of these spaces. LaterrnK.Venkateswarlu introduced the notion of direct sums in AbstractrnR-Vector spaces and established that every direct sum of AbstractrnR-Vector spaces has a basis provided each component has a basis.rnThis thesis is a further continuation on the theory of Abstract RVectorrnspaces. It is investigated by introducing special homomorphisms,rnstrong special homomorphisms, bilinear maps and fractionsrnin Abstract R-Vector Spaces. It is observed that special homomorphismrnis a normed Abstract R-Vector Space with a suitablernnorm. Certain properties regarding dual spaces has been obtainedrnlike some necessary and su_cient condition for two Abstract R-rnrnVector Spaces to be dual and some interesting results have beenrnproved on fractions in Abstract R-Vector Spaces.

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