On the Theory of Abstract R-Vector Spaces Over a Commutative Regular Ring
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Date
2016-01-01
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Addis Ababa University
Abstract
The 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.
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Keywords
Theory of Abstract, R-Vector Spaces, Over a Commutative, Regular Ring