The question is to show that for a ∈ R and u ∈ R, a unit, (a) = (ua), where (a) is the principal ideal defined by a, and same with (ua).
What I was thinking was that I could use the fact that (a) = {ar | r ∈ R} and (ua) = {uar | r ∈ R}, and ua ∈ R to show that they are equal.
Is this on the right track?
Yes your approach is on the right track. A standard trick to do these kind of problems is to show containment in both directions:
$(a) \subseteq (ua)$ and $(ua) \subseteq (a)$.
This will give the desired equality.