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.

  • $\begingroup$ Thank you for the reminder! $\endgroup$ – 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.

  • $\begingroup$ Okay brilliant, thanks! In general, are there many other common invariants for ring isomorphisms applicable to problems like this? $\endgroup$ – Wooster May 23 '14 at 9:53
  • $\begingroup$ whether or not the ring is a free module over its prime subring (of course, that wouldn't have helped here) $\;\;$ $\endgroup$ – user57159 May 23 '14 at 9:57
  • $\begingroup$ Two of the related questions don't have answers to your question per se, but do give a few ways you might show that rings aren't isomorphic: math.stackexchange.com/questions/296737/… has a lot of answers written by the same person, which you might find helpful. But to get some other perspectives you might see the three answers to math.stackexchange.com/questions/190252/… $\endgroup$ – Eric Stucky Jun 10 '14 at 17:24

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