Let R be a commutative ring with identity and let be the rel

Let R be a commutative ring with identity, and let be the relation on R defined by a b if and only if a = bu for some unit u R for each a, b R. Show that is an equivalence relation on R.

Solution

Let R be an integral domain.

if a ~ b , then a | b and b | a. These imply a = bu and b = av for some u,v R.

Then a = bu = ( av ) u

and so, a(1 - uv ) = 0.

Since R is an integral domain and a 0, we have

1 - uv = 0

thus, uv = 1

Since R is commutative , we have

v = u^-1

i.e.. u is the a unit.

Conversely,

let a = bu where u is a unit in R.

Then by definition,

b | a

also since u is a unit,

b = au^-1

which implies a | b

Therefore,

a b