Let $C_3 = \langle a | a^3 = e \rangle$ and let $R=(\mathbb{Z}/2)[C_3]$ be the group ring of $C_3$ with $\mathbb{Z}/2$ coefficients. Let $S = (\mathbb{Z}/2)[y]/(y^3-[1])$ and let $T = \mathbb{Z}[x]/(2,x^3-1)$. Prove that $S,T$, and $R$ are pairwise isomorphic rings.

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    $\begingroup$ Can you think of an isomorphism between $R$ and $S$ at least? If this is not possible, then try to write down the elements of $R$ and $S$ explicitly, and find a pattern that you can exploit. A similar sort of procedure should be adopted if you cannot figure out anything between $S$ and $T$. $\endgroup$ – астон вілла олоф мэллбэрг Nov 28 '18 at 3:23
  • $\begingroup$ For $S$ and $T,$ try to construct a surjective homomorphism from $\mathbb{Z}[x]$ to $S$ with kernel $(2, x^3-1).$ Again, the same for $R$ and $S.$ $\endgroup$ – dhk628 Nov 28 '18 at 5:45
  • $\begingroup$ From $(\mathbb{Z}/2)[x]$ to R it seems that an isomorphism would be the evaluation homomorphism at $a$. Would that work? $\endgroup$ – Wesley Nov 28 '18 at 16:35

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