Let $(R, +, \cdot)$ be a ring. $(S, +, \cdot)$ is a subring of $R$ if it is a ring itself and $S \subseteq R$.
Let $R$ be a ring. $S$ is a subring as soon as:
- \[1 \in S.\]
(Closure under subtraction) For all $a$, $b \in S$,
\[a - b \in S.\](Closure under multiplication) For all $a$, $b \in S$,
\[ab \in S.\]