Let $(R,+,\cdot,0,1)$ be a ring. $S$ is a subring as soon as
- $a-b\in S$ for all $a$, $b\in S$;
- $1\in S$;
- $ab\in S$ for all $a$, $b\in S$.
(I) guarantees that $(R,+,0)$ is a subgroup; (II) and (III) guarantee that $(R,\cdot,1)$ is a submonoid with the same neutral element.