Subring

From Wikipedia, the free encyclopedia

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 proper ideals become subrings (see below).

A subring of a ring (R, +, *) is 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.

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).

[edit] 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 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 unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity.