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	== Subring generated by a set ==		== Subring generated by a set ==
			{{see also\|Generator (mathematics)}}

			A special kind of subring of a ring {{mvar\|R}} is the subring '''generated by''' a subset {{mvar\|X}}, which is defined as the intersection of all subrings of {{mvar\|R}} containing {{mvar\|X}}.<ref>{{cite book \|last=Lovett \|first=Stephen \|date=2015 \|title=Abstract Algebra: Structures and Applications \|chapter=Rings \|pages=216-217 \|publisher=CRC Press \|publication-place=Boca Raton \|isbn=9781482248913}}</ref> The subring generated by {{mvar\|X}} is also the set of all ]s with integer coefficients of elements of ''X'', including the additive identity ("empty combination") and multiplicative identity ("empty product").{{cn\|date=August 2024}}
	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''. This subring 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'' ''']''' by ''X''. If ''S'' = ''R,'' we may say that the ring ''R'' is ''generated'' by ''X''.

			Any intersection of subrings of {{mvar\|R}} is itself a subring of {{mvar\|R}}; therefore, the subring generated by {{mvar\|X}} (denoted here as {{mvar\|S}}) is indeed a subring of {{mvar\|R}}. This subring {{mvar\|S}} is the smallest subring of {{mvar\|R}} containing {{mvar\|X}}; that is, if {{mvar\|T}} is any other subring of {{mvar\|R}} containing {{mvar\|X}}, then {{math\|''S'' ⊆ ''T''}}.
	The subring generated by ''X'' is the set of all ]s with integer coefficients of products of elements of ''X'' (including the empty linear combination, which is 0, and the empty product, which is 1).

			Since {{mvar\|R}} itself is a subring of {{mvar\|R}}, if {{mvar\|R}} is generated by {{mvar\|X}}, it is said that the ring {{mvar\|R}} is ''generated by'' {{mvar\|X}}.

	== See also ==		== See also ==

Revision as of 21:44, 8 August 2024

Subset of a ring that forms a ring itself

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Algebraic structure → Ring theory Ring theory
Basic conceptsRings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z}$ • Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebraCommutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$
Noncommutative algebraNoncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
v t e

Algebraic structure → Ring theory
Ring theory

Basic conceptsRings

• Subrings

• Ideal

• Quotient ring

• Fractional ideal

• Total ring of fractions

• Product of rings

• Free product of associative algebras

• Tensor product of algebras

Ring homomorphisms

• Kernel

• Inner automorphism

• Frobenius endomorphism

Algebraic structures

• Module

• Associative algebra

• Graded ring

• Involutive ring

• Category of rings

• Initial ring

\mathbb {Z}

• Terminal ring

0=\mathbb {Z} /1\mathbb {Z}

Related structures

• Field

• Finite field

• Non-associative ring

• Lie ring

• Jordan ring

• Semiring

• Semifield

Commutative algebraCommutative rings

• Integral domain

• Integrally closed domain

• GCD domain

• Unique factorization domain

• Principal ideal domain

• Euclidean domain

• Field

• Finite field

• Polynomial ring

• Formal power series ring

Algebraic number theory

• Algebraic number field

• Integers modulo n

• Ring of integers

• p-adic integers

\mathbb {Z} _{p}

• p-adic numbers

\mathbb {Q} _{p}

• Prüfer p-ring

\mathbb {Z} (p^{\infty })

Noncommutative algebraNoncommutative rings

• Division ring

• Semiprimitive ring

• Simple ring

• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra

Operator algebra

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 that shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.)

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).

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 $\mathbb {Z}$ and its quotients $\mathbb {Z} /n\mathbb {Z}$ have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to some ring $\mathbb {Z} /n\mathbb {Z}$ with n a nonnegative integer (see Characteristic). The integers $\mathbb {Z}$ correspond to n = 0 in this statement, since $\mathbb {Z}$ is isomorphic to $\mathbb {Z} /0\mathbb {Z}$ .

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

The ring of split-quaternions has subrings isomorphic to the rings of dual numbers, split-complex numbers and to the complex number field.

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.

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.

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 $\mathbb {Z}$ 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.

Ring extensions

Not to be confused with a ring-theoretic analog of a group extension.

If S is a subring of a ring R, then equivalently R is said to be a ring extension of S.

Subring generated by a set

Notes

^ Dummit & Foote 2004.
Lang 2002.
Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.

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.
Lang, Serge (2002). Algebra (3 ed.). New York. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)
Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.

Category:

Ring theory