Subring

SR#DEF. Subring.

Let $(R,+,\cdot)$ be a ring. $(S,+,\cdot)$ is a subring of $R$ if it is a ring and $S\subseteq R$.

SR#PROP-A.

Let $R$ be a ring. $S$ is a subring as soon as:

1. \[1\in S.\]

2. Closure under subtraction. For all $a$, $b\in S$,

   \[a-b \in S.\]

3. Closure under multiplication. For all $a$, $b\in S$,

   \[ab \in S.\]