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