# Showing that three rings are isomorphic

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.

• 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 $T$. – астон вілла олоф мэллбэрг Nov 28 '18 at 3:23
• 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 $S.$ – dhk628 Nov 28 '18 at 5:45
• From $(\mathbb{Z}/2)[x]$ to R it seems that an isomorphism would be the evaluation homomorphism at $a$. Would that work? – Wesley Nov 28 '18 at 16:35