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.

Proper Subring
A subring $S$ of a ring $R$ is a proper subring iff $S$ is not the Null Ring nor $R$ itself.

Also see

 * Subring Test
 * Null Ring and Ring Itself Subrings