**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

.)