Show that the ring $3\mathbb Z$ is not isomorphic to the ring $5\mathbb Z$.
I see that they are not but I am not sure how to go about proving it. We went over a similar problem, disproving it by using that the number of units in the rings were not the same but that doesn't seem to apply in this case.
They are not isomorphic, to see this, note that $3+3+3=3^2$, thus $3\mathbb{Z}$ has a non-zero element $x$ such that $3x=x^2$. There is no such element in $5\mathbb{Z}$.