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.
2026-04-23 11:10:34.1776942634
Showing that two rings are not isomorphic
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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.