1. ## Principal ideal

Let Z be the integers and let I={5k| k=0,±1,±2,⋯} be the principal ideal generated by 5. Let a,b∈Z, show that a≡b (mod 5) if and only if a-b∈I.

Need help starting this problem. do i show that I is a subset of Z?

Because i know that a nonempty subset I of a ring R is an ideal iff it has these properties:
i) if a,b are elements of I, then a-b is an element of I;
ii) if r∈ R and a ∈ I, then ra ∈ I and ar ∈ I.

2. Originally Posted by empressA88
Let Z be the integers and let I={5k| k=0,±1,±2,⋯} be the principal ideal generated by 5. Let a,b∈Z, show that a≡b (mod 5) if and only if a-b∈I.

Need help starting this problem. do i show that I is a subset of Z?

Because i know that a nonempty subset I of a ring R is an ideal iff it has these properties:
i) if a,b are elements of I, then a-b is an element of I;
ii) if r∈ R and a ∈ I, then ra ∈ I and ar ∈ I.
$a\equiv b\bmod{5} \iff 5\mid a-b \iff a-b=5k\iff a-b\in I$