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subring
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{{No footnotes|date=November 2018}}In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

Formal definition

A subring of a ring {{nowrap|(R, +, ∗, 0, 1)}} is a subset S of R that preserves the structure of the ring, i.e. a ring {{nowrap|(S, +, ∗, 0, 1)}} with {{nowrap|S ⊆ R}}. Equivalently, it is both a subgroup of {{nowrap|(R, +, 0)}} and a submonoid of {{nowrap|(R, ∗, 1)}}.

Examples

The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring.Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to {{nowrap|1=n = 0}} in this statement, since Z is isomorphic to Z/0Z.

Subring test

The subring test is a theorem that states that for any ring R, a subset S of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.As an 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].

Ring extensions

Not to be confused with a ring-theoretic analog of a group extension. For that, see Ring extension.
If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/S in similar notation to that for field extensions.

Subring generated by a set

Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

Relation to ideals

Proper ideals are subrings that are closed under both left and right multiplication by elements from R.If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, 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 unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
  • The proper ideals of Z have no multiplicative identity.
If I is a prime ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime in S. In this case one says that I lies over I âˆ© S. The situation is more complicated when R is not commutative.

Profile by commutative subrings

A ring may be profiled{{clarify|what "profile" means here?|date=June 2016}} by the variety of commutative subrings that it hosts:

See also

References

  • BOOK, Iain T. Adamson, Elementary rings and modules, University Mathematical Texts, Oliver and Boyd, 1972, 0-05-002192-3, 14–16,
  • Page 84 of BOOK, David Sharpe, Rings and factorization, Cambridge University Press, 1987, 0-521-33718-6, 15–17,


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