Definition:Subring

Definition
Let $\left({R, +, \circ}\right)$ be an algebraic structure with two operations.

A subring of $\left({R, +, \circ}\right)$ is a subset $S$ of $R$ such that $\left({S, +_S, \circ_S}\right)$ is a ring.

Also defined as
Sources which deal only with rings with unity typically demand that the unity is also part of a subring.

Some sources insist that $R$ must be a ring for $S$ to be definable as a subring, but this limitation is unnecessarily restricting.

Also see

 * Subring Test
 * Null Ring and Ring Itself Subrings