module homomorphism, ring, subset

**Background:** Let be a ring, a multiplicatively closed subset, and let , be modules. Let be an module homomorphism, and let denote the module homomorphism, and let You may assume without proof that is a well-defined module homomorphism.

**Prove:**

(a) Prove that if is injective, then is injective.

(b) Prove that if is surjective, then is surjective.

(c) Prove that if is a finitely generated free module with basis then is a finitely generated free module with basis

(d) Now suppose is an integral domain and that , are positive integers. Prove that if , then . (Hint: consider .)