# User Contributed Dictionary

### Noun

### 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:- The quaternion ring H contains only the complex plane as a planar subring
- The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane
- The ring of 3 x 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 x 3 matrices.

subring in Korean: 부분환

subring in Polish: Podpierścień

subring in Russian: Подкольцо