An ideal $I$ of a ring $R$ must satisfy: (i) $(I,+)$ is a subgroup, and (ii) for all $r\in R$, $a\in I$: $ra\in I$ and $ar\in I$. In $\mathbb{R}$ (a field), the only ideals are $\{0\}$ and $\mathbb{R}$ itself. $\{0\}$ is the trivial ideal. $\mathbb{Z}$ and $\mathbb{Q}$ are subrings but not ideals (e.g. $\pi\cdot 1 = \pi \notin\mathbb{Z}$).