AskDefine | Define subring

User Contributed Dictionary

English

Noun

  1. a ring which is contained in a larger ring, such that the multiplication and addition on the former are a restriction of those on the latter

Usage notes

If the larger ring contains a multiplicative identity, certain authors also require that the identity also be contained in the subring.

Extensive Definition

In mathematics, a subring is a subset of a ring, which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity (if it exists). This leads to the added advantage that ideals become subrings (see below).
A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the mutiplicative 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 (with multiplicative identity) other than itself.
Every ring has a unique smallest subring, isomorphic to either the integers Z or some ring Z/nZ with n a nonnegative integer (see characteristic).
The subring test states that for any ring, a nonempty subset of that ring is itself a ring if it is closed under multiplication and subtraction, and has a multiplicative identity.

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 never subrings since if they contain the identity then they must be the entire ring. For example, ideals in Z are of the form nZ where n is any integer. These are subrings if and only if n = ±1 (otherwise they do not contain 1) in which case they are all of Z.
If one omits the requirement that rings have a unit element, then subrings need only contain 0 and be closed under addition, subtraction 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 = of the ring Z × 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.

Profile by commutative subrings

A ring may be profiled by the variety of commutative subrings that it hosts:
subring in Korean: 부분환
subring in Polish: Podpierścień
subring in Russian: Подкольцо
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