Teshome, Zelalem (PhD)Wondie, Litegebe2020-10-282023-11-042020-10-282023-11-042016-01-01http://etd.aau.edu.et/handle/123456789/22969The notion of an Abstract Boolean Vector space (an Abstract BVector space) is introduced by Subrahmanyam N.V. and he studied intensively on this spaces. Later N.Raja Gopala Rao introduced the concept of an Abstract R-Vector spaces as a generalization of Abstract Boolean Vector space of Subrahmanyam N.V. He introduced the notion of linear endomorphisms and a_ne transformations in Abstract R-Vector spaces and studied its properties. Further, he made a study on the geometric aspect of these spaces. Later K.Venkateswarlu introduced the notion of direct sums in Abstract R-Vector spaces and established that every direct sum of Abstract R-Vector spaces has a basis provided each component has a basis. This thesis is a further continuation on the theory of Abstract RVector spaces. It is investigated by introducing special homomorphisms, strong special homomorphisms, bilinear maps and fractions in Abstract R-Vector Spaces. It is observed that special homomorphism is a normed Abstract R-Vector Space with a suitable norm. Certain properties regarding dual spaces has been obtained like some necessary and su_cient condition for two Abstract R- Vector Spaces to be dual and some interesting results have been proved on fractions in Abstract R-Vector Spaces.enTheory of AbstractR-Vector SpacesOver a CommutativeRegular RingThesis