Subring

Short description: Subset of a ring that forms a ring itself

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

• Fractional ideal
• Total quotient ring
• Product ring
• Free product of associative algebras
• Tensor product of rings

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings

• Initial ring [math]\displaystyle{ \mathbb{Z} }[/math]
• Terminal ring [math]\displaystyle{ 0 = \mathbb{Z}_1 }[/math]

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

* * *

Commutative algebra

Commutative rings

• Integral domain

• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

• Finite field
• Composition ring

• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit
• Zero ring [math]\displaystyle{ \mathbb{Z}_1 }[/math]
• Integers modulo pn [math]\displaystyle{ \mathbb{Z}/p^n\mathbb{Z} }[/math]
• Prüfer p-ring [math]\displaystyle{ \mathbb{Z}(p^\infty) }[/math]
• Base-p circle ring [math]\displaystyle{ \mathbb{T} }[/math]
• Base-p integers [math]\displaystyle{ \mathbb{Z} }[/math]
• p-adic rationals [math]\displaystyle{ \mathbb{Z}[1/p] }[/math]
• Base-p real numbers [math]\displaystyle{ \mathbb{R} }[/math]
• p-adic integers [math]\displaystyle{ \mathbb{Z}_p }[/math]
• p-adic numbers [math]\displaystyle{ \mathbb{Q}_p }[/math]
• p-adic solenoid [math]\displaystyle{ \mathbb{T}_p }[/math]

Algebraic geometry

• Affine variety

* * *

Noncommutative algebra

Noncommutative rings

• Division ring
• Semiprimitive ring
• Simple ring
• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra
Operator algebra

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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.

## Contents

* 1 Definition
* 2 Examples
* 3 Subring test
* 4 Center
* 5 Prime subring
* 6 Ring extensions
* 7 Subring generated by a set
* 8 See also
* 9 Notes
* 10 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).

## 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. ↑ 1.0 1.1 Dummit & Foote 2004.
2. ↑ 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.

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