Eisenstein integers and $\mathbb{Z}C_3$

Question

The Eisenstein Integers $a+b\omega$ with norm $N(x)=a^2-ab+b^2$ form a commutative ring, as does the group ring $\mathbb{Z}C_3=\{\sum_{g\in C_3} a_g g \mid g \in C_3, a_g \in \mathbb{Z}\}$. $\mathbb{Z}C_3$ also has a multiplicative map: $M(x)=a^2+b^2+c^2-ab-bc-ac$, where $a,b,c$ are the coefficients of an element in $\mathbb{Z}C_3$.

Are the Eisenstein integers related (isomorphic?) to the augmentation ideal $\{\sum_{g\in C_3} a_g g \mid g \in C_3, a_g \in \mathbb{Z}, \sum_{g \in C_3} a_g = 0\}$ or a related subset, maybe of $\mathbb{Q}C_3$, where $M$ plays the role of $N$?

Answer 1

The answer is certainly affirmative, if you extend the coefficients to $\Bbb{Q}$. This is because the group algebra $\Bbb{Q}C_3\cong \Bbb{Q}[x]/\langle x^3-1\rangle$. Here $x^3-1=(x-1)(x^2+x+1)$ factors into coprime irreducible factors, so by the Chinese Remainder Theorem we have $$ \Bbb{Q}[x]/\langle x^3-1\rangle\cong \Bbb{Q}[x]/\langle x-1\rangle\oplus\Bbb{Q}[x]/\langle x^2+x+1\rangle. $$ The latter summand is obviously isomorphic to the field $\Bbb{Q}[\omega]$. The restriction of the isomorphism given by CRT to the augmentation ideal, $I=(x-1)\left(\Bbb{Q}[x]/\langle x^3-1\rangle\right)$, maps it isomorphically onto the latter summand.

I expect some potential problems when you restrict the coefficients to integers. Largely because I don't see how to write $1$ as a $\Bbb{Z}[x]$-linear combination of $x-1$ and $x^2+x+1$. I only get $3=(x^2+x+1)-(x+2)(x-1)$. I dare not say anything definite right away.

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Edit:

If we denote the generator of $C_3$ by $c$, the mapping $$ a_1\cdot1+a_2\cdot c+a_3\cdot c^2\mapsto a_1+a_2\omega+a_3\omega^2 $$ for all integers $a_1,a_2,a_3$ does map the group ring $\Bbb{Z}C_3$ homomorphically to the ring of Eisensteinian integers. The augmentation ideal is additively generated by $1-c$ and $c-c^2$. The images of these elements $1-\omega$ and $\omega-\omega^2=2\omega-1$. These two numbers additively generate the prime ideal of $\Bbb{Z}[\omega]$ lying above the ramified prime $3$. So the image of the augmentation ideal is of index three.