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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 1 and is closed under multiplication.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some modular arithmetic Zn.