# Subring

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

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 (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**×**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* ∩ *S*. The situation is more complicated when *R* is not commutative.

## 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 × 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 × 3 matrices.

## References

- Iain T. Adamson (1972).
*Elementary rings and modules*. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. - Page 84 of Lang, Serge (1993),
*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 - David Sharpe (1987).
*Rings and factorization*. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.