# Showing that two rings are not isomorphic

I have two rings: $R_1 = \mathbb{Z}_2[x]/\langle x^4+1\rangle$ and $R_2 = \mathbb{Z}_4[x]/\langle x^2+1\rangle$. I've shown that these have the same number of elements. Now I am struggling to show that they are not isomorphic. In general, are there any invariants in rings to check? As in groups, where we could check that the order of elements under an isomorphism is the same etc.

• Thank you for the reminder! – Wooster Jun 10 '14 at 20:39

One invariant of a ring is its characteristic, the smallest natural number $n$ such that $$\underbrace{1+\ldots+1}_{\text{n times}}=0$$ The two rings you are considering have different characteristics.
• whether or not the ring is a free module over its prime subring (of course, that wouldn't have helped here) $\;\;$ – user57159 May 23 '14 at 9:57