Subject: Grbner Bases for Modules Let R = k[x1, .., Xn] be a polynomial ring over a field k. you extend the definition of Grbner basis to the module Rm. You can use the following references 1)Cox, David A, Little, John, Oshea, Donal, Using Geometry, Chapter 5. 2) William W. Adams, Philippe Loustaunau , An Introduction to Grbner Bases, Chapter 3. You should do the followings : 1) Define monomials in Rm. 2) Define TOP and POT order on monomials. Give some examples. 3) Define divisibility of monomials and then define division algorithm in Rm. 4) Give some examples of divisions using TOP and POT orders. 5) Define the Grbner basis in R. 6) Define S-polynomial in Rm and gives some examples of s-polynomials. 7) Explain the Buchberger Algorithm for Grbner Basis computation in Rm. 8) Apply Bucberger algorithm to compute a Grbner basis a module.

Instructions 1) You can take the definitions from source books. Be careful they use different notations. The notations you will use throughout your project must be the consistent. 2) You cannot use the examples in the source books in your project. The examples you give should be different from both the source books

also we use SINGULAR application to calculate Grobner basis