RINGS OVER WHICH A FREE MODULE IS GENERATED FROM FLAT MODULE

The aim of this dissertation is to determine the rings over which a free module is generated from flat module. Modules are not always like vector spaces, Dylan (2010); therefore the purpose of this dissertation is to generate a module that behaves so much like vector spaces, that is a module with bases which are linearly independent. Over any rings, projective and flat modules are generated from free module but the reverse is not always true, Konrad (2012); rather, there are specific rings that need to be identified in order to generate free module from flat module. From the well-known fact that over a local ring a finitely generated flat module is free; therefore in order to generate free module from flat module, the elements of the local ring were mapped with the elements of an abelian group using the four module axioms. This dissertation considered rational number as a -module (which is also flat module), an induction argument showed that any number of elements greater than two in are linearly dependent and cannot form basis for , hence -module are not free. The dissertation also considered the ring of rational numbers with even numerator and odd denominator as a local ring whose maximal ideal is 2 over module (flat module); the result showed that, ring of rational numbers with even numerator and odd denominator formed basis for rational number and the ring of rational numbers is a subset of . An ideal was also identified as a subset of and the result showed that the set of generators formed basis for and is the principal ideal generated by single element subset of . Therefore, -module over ring of rational numbers with odd denominator and -module over an ideal have generating sets consisting of linearly independent elements. The ring of rational numbers and ideal formed the rings over which the free module is generated from flat module. Finally, a description of the Smith Normal Form algorithm and related examples were given; this provides ability to compute bases for -modules and these bases are linearly independent; also the modules generated are free.