Subring

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In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations.

More precisely, given a ring (R, +, *), we say that a subset S of R is a subring of R if it is a ring under the restriction of + and * to S, and contains the same multiplicitive identity 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 no subrings other than itself. Note that ideals in Z, which are of the form nZ, where n is any integer, are not subrings (unless n = ±1) as they do not contain 1. In general, a proper ideal is never a subring since if it contains the identity then it must be the entire ring.

If one omits the requirement that rings have a unit element, then subrings need only be closed under addition and multiplication and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

  • The ideal I = {(z,0)|z in Z} of the ring Z × Z = {(x,y)|x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unit, and a "subring-without-unit", but not a "subring-with-unit" of Z × Z.
  • The proper ideals of Z have no multiplicative identity.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some cyclic group Z/nZ (see characteristic).

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