Bekele, Berhanu(PhD)Belete, Solomon2018-07-182023-11-042018-07-182023-11-042012-06http://etd.aau.edu.et/handle/123456789/9136In this thesis we present the basic concepts and results of Groebner basis for a polynomial ideal over a field and introduce an algorithm for computing and then present an improvement of Buchberger's algorithms for computing Groebner basis by reducing number of S-polynomials without computing them. This paper deals with Groebner basis for a polynomial ideal over a ring by defining the module of a solution of a homogenous linear equation with polynomial coeffcients (called the syzygy module). And finally, we will see the application of Groebner basis in detail.enGroebner Basis and its ApplicationsThesis