Subring

In mathematics, a subring of a ring R is a subset of R that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as R.[lower-alpha 1]

## Contents

* 1 Definition
* 1.1 Variations
* 2 Examples
* 3 Subring generated by a set
* 4 Ring extension
* 4.1 Adjoining
* 4.2 Prime subring
* 5 See also
* 6 Notes
* 7 References
* 7.1 General references

## 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).

Equivalently, S is a subring if and only if it contains the multiplicative identity of R, and is closed under multiplication and subtraction. This is sometimes known as the subring test.[1]

### Variations

Some mathematicians define rings without requiring the existence of a multiplicative identity (see Ring (mathematics) § History). In this case, a subring of R is a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of R. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of R that is a subring of R is R itself.

## Examples

* The ring of integers ℤ is a subring of both the field of real numbers and the polynomial ring ℤ[X].[1]

* ℤ and its quotients ℤ/nℤ have no subrings (with multiplicative identity) other than the full ring.[1]

* Every ring has a unique smallest subring, isomorphic to some ring ℤ/nℤ with n a nonnegative integer (see Characteristic). The integers ℤ correspond to n = 0 in this statement, since ℤ is isomorphic to ℤ/0ℤ.[2]

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

## Subring generated by a set

See also: Generator (mathematics)

A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X.[3] The subring generated by X is also the set of all linear combinations with integer coefficients of products of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").[4]

Any intersection of subrings of R is itself a subring of R; therefore, the subring generated by X (denoted here as S) is indeed a subring of R. This subring S is the smallest subring of R containing X; that is, if T is any other subring of R containing X, then S ⊆ T.

Since R itself is a subring of R, if R is generated by X, it is said that the ring R is generated by X.

## Ring extension

Subrings generalize some aspects of field extensions. If S is a subring of a ring R, then equivalently R is said to be a ring extension[lower-alpha 2] of S.

### Adjoining

If A is a ring and T is a subring of A generated by R ∪ S, where R is a subring, then T is a ring extension and is said to be S adjoined to R, denoted R[S]. Individual elements can also be adjoined to a subring, denoted R[a1, a2, ..., an].[5][3]

For example, the ring of Gaussian integers ℤ[i] is a subring of ℂ generated by ℤ∪{i}, and thus is the adjunction of the imaginary unit i to ℤ.[3]

### 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 ℤ 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.

## See also

* Integral extension
* Group extension
* Algebraic extension
* Ore extension

## Notes

1. ↑ In general, not all subsets of a ring R are rings.
2. ↑ Not to be confused with the ring-theoretic analog of a group extension.

## References

1. ↑ 1.0 1.1 1.2 Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9. https://archive.org/details/abstractalgebra0000dumm_k3c6.
2. ↑ Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.
3. ↑ 3.0 3.1 3.2 Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.
4. ↑ Robinson, Derek J. S. (2022). Abstract Algebra: An Introduction with Applications (3rd ed.). Walter de Gruyter GmbH & Co KG. p. 109. ISBN 9783110691160. https://books.google.com/books?id=DOxcEAAAQBAJ&pg=PA109.
5. ↑ Gouvêa, Fernando Q. (2012). "Rings and Modules". A Guide to Groups, Rings, and Fields. Washington, DC: Mathematical Association of America. p. 145. ISBN 9780883853559.

### General references

* Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
* Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6. https://archive.org/details/ringsfactorizati0000shar.

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