Let R be a commutative ring with identity.?

Question

Define a relation ~ on R by saying a ~ b if a = bu for some u (element of R). Show that ~ is an equivalence relation.

Answer 1

a = ae where e is identity. so aRa so R is reflexive.

If aRb the a = bu and so b = a(u^ -1) = a( u inverse), so bRa so R is symmetric

Id aRb abd bRc the a = bu and b = cv so a = c(vu), so aRc so R is transitive.

Hence its equivalence.

Answer 2

bskelkar's proof is correct, but your question is wrong.  You need that u is a unit in R.

Steve