Quotient polynomial rings over the infinite field containing Q are involved. The computations concern the univariate polynomial rings and the multivariate polynomial rings in two variables. In both cases a vector space basis for the quotient is constructed. The case of of several variables includes cases of infinite dimensions.We shall restrict ourselves to the finite computations. Ring operations of addition and multiplication on the quotients are computed as well. Goebner basis is used and the computations are carried out in Maple 13.