Concerning subrings of a integral group ring

Question

Let $G$ be a group .

Does a subring of the integral group ring $\mathbb{Z}[G]$ has the form $\mathbb{Z}[H]$ for a subgroup $H$ of $G$?

Thanks in advance.

Answer 1

Not necessarily.

Let $G=\{e,g\}$ have two elements. Consider, say $$R=\{ae+bg:a,b\in\Bbb Z:a-b\in2\Bbb Z\}.$$