I would like to know the automorphism groups of the rational integers $\mathbf Z$, the Gaussian integers $\mathbf Z[i]$, and the Eisenstein integers $\mathbf Z[\omega]$.
My question is, would $\text{Aut}(R)$ refer to the group of group automorphisms or to the group of ring automorphisms?
If $\text{Aut($R$)}$ refers to the group of group automorphisms, I would guess $\text{Aut}(\mathbf Z) = C_2$, $\text{Aut}(\mathbf Z[i]) = C_4$, and $\text{Aut}(\mathbf Z[\omega]) = C_6$.
If $\text{Aut($R$)}$ refers to the group of ring automorphisms, I would guess $\text{Aut}(\mathbf Z) = C_1$, $\text{Aut}(\mathbf Z[i]) = C_2$, and $\text{Aut}(\mathbf Z[\omega]) = C_2$.
Also, what if I wanted to compute the automorphism group of a field?