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subring

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**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*, +, âˆ—, 0, 1)}} is a subset

*S*of

*R*that preserves the structure of the ring, i.e. a ring {{nowrap|(

*S*, +, âˆ—, 0, 1)}} with {{nowrap|

*S*âŠ†

*R*}}. Equivalently, it is both a subgroup of {{nowrap|(

*R*, +, 0)}} and a submonoid of {{nowrap|(

*R*, âˆ—, 1)}}.

## Examples

The ring**Z**and its quotients

**Z**/

*n*

**Z**have no subrings (with multiplicative identity) other than the full ring.Every ring has a unique smallest subring, isomorphic to some ring

**Z**/

*n*

**Z**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**/0

**Z**.

## 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.*

*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 (without unity) 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**Ã—**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**Ã—**Z**. - The proper ideals of
**Z**have no multiplicative identity.

*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:- 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 Ã— 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent Îµ of order 3 (ÎµÎµÎµ = 0 â‰ ÎµÎµ). 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 Ã— 3 matrices.

## See also

## References

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

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