M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring homomorphism if Kerf={0} prove that f is one to one. (5 Marks)

Prove the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.

Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]

Prove Theorem If and are relatively prime polynomials over the field and if in , then in .