Subrings

Definition

Let $(R, +, \cdot )$ be a ring. A ring $(S, +, \cdot )$ is a subring of $(R, +, \cdot )$ if $S \subset R$.

Verification

If $(R, +, \cdot )$ and $S \subset R$, then $+$ is associative and commutative on $S$ because it is on $R$. Likewise $\cdot $ is associative on $S$ and $+$ and $\cdot $ distribute over each on $S$ because they do on $R$. So we have restricted the number of conditions to check, and arrive at our first statement of sufficient conditions on $S$ that ensure $(S, +, \cdot )$ is a ring.