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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 the identity and is closed under multiplication.
For example, the ring Z of integers is a subring of the field 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 Zn.
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