(iii) If a,b are not unit, then a+ b is not a unit.

3. [20%] Prove the following simple form of the structure theorem for finitely generated mod- ules over a principal ideal domain (so you ca can not apply the struc cture theo orem directly). Let A be a principal ideal domain and M a 2 x 2 matrix whose entries are in A. Show that there exist invertible matrices P, Q with entries in A and a,β A with a | β such that