Let $m\mathbb{Z}$ be an ideal of the ring $2\mathbb{Z}.$ Is it true that for any $m>2, 2\mathbb{Z}/m\mathbb{Z}=(2)$ modulo $m$? Where $m\mathbb{Z}$ is an ideal of $2\mathbb{Z}$ and $(2):=2\mathbb{Z}$.
I feel that it is yes, using the following examples. Let $2\mathbb{Z}$ be the ring of even integers. Consider the ideals $4\mathbb{Z}$ and $8\mathbb{Z}$ of $2\mathbb{Z}$. From $2\mathbb{Z}/4\mathbb{Z}=\{\bar{0},\bar{2}\}=(2)$ modulo $4$ and $2\mathbb{Z}/8\mathbb{Z}=\{\bar{0},\bar{2},\bar{4},\bar{6}\}=(2)$ modulo $8$.