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Subring

In abstract algebra, a branch of mathematics, a subring is a part of a ring, which is a ring itself.

More precisely, 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 has subrings of the form nZ, where n is any integer.

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

03-10-2013 05:06:04