A set A of integers is called an ideal if and only ifi. 0 ∈ A,ii. if a ∈ A, then also −a ∈...

A set A of integers is called an ideal if and only if

i. 0 ∈ A,

ii. if a ∈ A, then also −a ∈ A, and

iii. il a, b ∈ A, then a + b ∈ A.

[BB] For any integer n ≥ 0, recall than Z = {kn | k ∈ Z} denotes the set of multiples of n.

(a) Prove that nZ is an ideal of the integers.

(b) Let A be any ideal of Z. Prove that A = nZ for some n ≥ 0 by establishing each of the following statements.

i. If A contains only one element, then A is of the desired form.

Now assume that A contains more than one element.

ii. Show that A contains a positive number.

iii. Show that A contains a smallest positive number n.

iv. nZ ⊆ A, where n is the integer found in iii.

v. A ⊆ nZ. [Hint: 4.1.5, the Division Algorithm.]