Given a ring (

*R*, +, *), we say that a subset

*S*of

*R*is a

**subring**thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as

*R*. A subring is just a subgroup of (

*R*, +) which contains 1 and is closed under multiplication.

For example, the ring **Z** of integers is a subring of the field (mathematics) of real numbers and also a subring of the ring of polynomials **Z**[*X*]. The ring **Z** itself doesn't have any subrings except itself.

Every ring has a unique smallest subring, isomorphic to either the integers **Z** or some modular arithmetic **Z**_{n}.