Abstract Algebra

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Groups and Rings: Extension to a Ring of a Group Isomorphism


Let $\mathbb{Z}$ be the ring of integers, and let $\omega$ be a complex number with $\omega^2+\omega+1=0$.

Define rings $R_1$, $R_2$, $R_3$ by $R_1=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}, \quad R_2=\mathbb{Z}[\omega],\quad R_3=\mathbb{Z}[x]/(x^3),$
and let $G_1$, $G_2$, $G_3$ be their respective underlying additive groups.

(The ring $R_2=\mathbb{Z}[\omega]$ is called the ring of Eisenstein integers.)

1) The pair $G_i, G_j$ of groups, with $i\not=j$, are


2) The pair $R_i,R_j$ of rings, with $i\not=j$, are