Let
R be a
commutative integral domain. Elements
a,b of
R
are called
associates if there exists a
unit u in
R
such that
a=bu
Note that this happens iff a divides b and b
divides a.
Proof If a=bu, for some unit u then by definition
b divides a. Since b=au-1 we also see that
a divides b.
On the other hand if we have a=bu and b=av, for some
u,v elements of R then substituting we see that
a=avu, that is a(1-vu)=0. Since R is an integral
domain we deduce that a and b are associates.
For example, 3 and -3 are associate integers.