I am reading Dummit and Foote on Ring Theory - ch 7

On page 228 D&F give examples of subrings

For example 2 they write

"$\displaystyle 2 \mathbb{Z} $ is a sub-ring of $\displaystyle \mathbb{Z} $ as is $\displaystyle n \mathbb{Z} $ for any integer n.

The ring $\displaystyle \mathbb{Z} / n \mathbb{Z} $ is not a subring of $\displaystyle \mathbb{Z} $ for any $\displaystyle n \geq 2 $ "

My question is surely the ring $\displaystyle \mathbb{Z} / n \mathbb{Z} $a subring of $\displaystyle \mathbb{Z} $ if we identify the equivalence class $\displaystyle \overline{0}$ with 0 and $\displaystyle \overline{1} $ with 1is

Perhaps the point D&F are making is that 0 and 1 are not the same as $\displaystyle \overline{0}$ and $\displaystyle \overline{1} $?

Can someone help by clarifying this issue

Peter