# Subring

__: Subset of a ring that forms a ring itself__

**Short description**

Algebraic structure → Ring theoryRing theory |
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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*. (Note that a subset of a ring *R* need not be a ring.) 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.

## 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 [math]\displaystyle{ \mathbb{Z} }[/math] and its quotients [math]\displaystyle{ \mathbb{Z}/n\mathbb{Z} }[/math] have no subrings (with multiplicative identity) other than the full ring.^{[1]}^{:228}

Every ring has a unique smallest subring, isomorphic to some ring [math]\displaystyle{ \mathbb{Z}/n\mathbb{Z} }[/math] with *n* a nonnegative integer (see *Characteristic*). The integers [math]\displaystyle{ \mathbb{Z} }[/math] correspond to *n* = 0 in this statement, since [math]\displaystyle{ \mathbb{Z} }[/math] is isomorphic to [math]\displaystyle{ \mathbb{Z}/0\mathbb{Z} }[/math].^{[2]}^{:89-90}

## Subring test

The **subring test** is a theorem that states that for any ring *R*, a subset *S* of *R* is a subring if and only if it contains the multiplicative identity of *R*, and is closed under multiplication and subtraction.^{[1]}^{:228}

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*].

## Center

The center of a ring is the set of the elements of the ring that commute with every other element of the ring. That is, x belongs to the center of the ring R if [math]\displaystyle{ xy=yx }[/math] for every [math]\displaystyle{ y\in x. }[/math]

The center of a ring R is a subring of R, and R is an associative algebra over its center.

## Prime subring

The intersection of all subrings of a ring R is a subring that may be called the *prime subring* of R by analogy with prime fields.

The prime subring of a ring R is a subring of the center of R, which is isomorphic either to the ring [math]\displaystyle{ \Z }[/math] of the integers or to the ring of the integers modulo n, where n is the smallest positive integer such that the sum of n copies of 1 equals 0.

## Ring extensions

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*.

## See also

- Integral extension
- Group extension
- Algebraic extension
- Ore extension

## Notes

- ↑
^{1.0}^{1.1}Dummit & Foote 2004. - ↑ Lang 2002.

## References

- Adamson, Iain T. (1972).
*Elementary rings and modules*. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. - Dummit, David Steven; Foote, Richard Martin (2004).
*Abstract algebra*(Third ed.). Hoboken, NJ: John Wiley & Sons. ISBN 0-471-43334-9. https://archive.org/details/abstractalgebra0000dumm_k3c6. - Lang, Serge (2002).
*Algebra*(3 ed.). New York. ISBN 978-0387953854. https://archive.org/details/algebra-serge-lang. - Sharpe, David (1987).
*Rings and factorization*.*Cambridge University Press*. pp. 15–17. ISBN 0-521-33718-6. https://archive.org/details/ringsfactorizati0000shar.

Original source: https://en.wikipedia.org/wiki/Subring.
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