Let R be a commutative ring with identity.
Show that the relation "is an associate of" is an equivalence relation on R.
I know that if R is a commutative ring with identity, then b is an associate of a if b = au for some unit u $\in$ R and for some a, b $\in$ R.
To show something is an equivalence relation is to show transitivity, symmetry, and reflexivity.
Reflexivity: As R is a commutative ring with identity, suppose a, b $\in$ R. Then (1) a = b and b = a. Hence, for some unit u $\in$ R, by the definition of an associate, b = au and so a = bu by (1).
I am stuck on both transitivity and associativity. Any feedback would be much appreciated!
Symmetrty: if $b=au$ for some unit $u$, then $a=bu^{-1}$ and $u^{-1}$ is a unit.
Transitivity: if $b=au$ and $c=bv$ for some units $u,v$, then $c=bv=auv=aw,$ where $w=uv$ is a unit.