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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 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.

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 $\mathbb {Z}$ 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 $\mathbb {Z} /0\mathbb {Z}$ .

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 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.

Ring extensions

Not to be confused with a ring-theoretic analog of a group extension. For that meaning, see an old version of the article Idealization of a module.

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 of 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.

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

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

Revision as of 18:49, 19 April 2023 editTMM53 (talk \| contribs)Extended confirmed users841 editsNo edit summary← Previous edit		Revision as of 01:47, 7 November 2023 edit undoQuondum (talk \| contribs)Extended confirmed users37,014 edits italic reference: this is to the article, not to its subject matterNext edit →
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	In ], a '''subring''' of ''R'' is a ] of a ] that is itself a ring when ]s of addition and multiplication on ''R'' are restricted to the subset, and which shares the same ] 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 ]s 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.		In ], a '''subring''' of ''R'' is a ] of a ] that is itself a ring when ]s of addition and multiplication on ''R'' are restricted to the subset, and which shares the same ] 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 ]s 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==		== Definition ==
	A subring of a ring {{nowrap\|(''R'', +, ∗, 0, 1)}} is a subset ''S'' of ''R'' that preserves the structure of the ring, i.e. a ring {{nowrap\|(''S'', +, ∗, 0, 1)}} with {{nowrap\|''S'' ⊆ ''R''}}. Equivalently, it is both a ] of {{nowrap\|(''R'', +, 0)}} and a ] of {{nowrap\|(''R'', ∗, 1)}}.		A subring of a ring {{nowrap\|(''R'', +, ∗, 0, 1)}} is a subset ''S'' of ''R'' that preserves the structure of the ring, i.e. a ring {{nowrap\|(''S'', +, ∗, 0, 1)}} with {{nowrap\|''S'' ⊆ ''R''}}. Equivalently, it is both a ] of {{nowrap\|(''R'', +, 0)}} and a ] of {{nowrap\|(''R'', ∗, 1)}}.

	==Examples==		== Examples ==
	The ring <math>\mathbb{Z}</math> and its quotients <math>\mathbb{Z}/n\mathbb{Z}</math> have no subrings (with multiplicative identity) other than the full ring.{{sfn\|Dummit\|Foote\|2004}}{{rp\|228}}		The ring <math>\mathbb{Z}</math> and its quotients <math>\mathbb{Z}/n\mathbb{Z}</math> have no subrings (with multiplicative identity) other than the full ring.{{sfn\|Dummit\|Foote\|2004}}{{rp\|228}}

	Every ring has a unique smallest subring, isomorphic to some ring <math>\mathbb{Z}/n\mathbb{Z}</math> with ''n'' a nonnegative integer (see ]). The integers <math>\mathbb{Z}</math> correspond to {{nowrap\|1=''n'' = 0}} in this statement, since <math>\mathbb{Z}</math> is isomorphic to <math>\mathbb{Z}/0\mathbb{Z}</math>.{{sfn\|Lang\|2002}}{{rp\|89-90}}		Every ring has a unique smallest subring, isomorphic to some ring <math>\mathbb{Z}/n\mathbb{Z}</math> with ''n'' a nonnegative integer (see '']''). The integers <math>\mathbb{Z}</math> correspond to {{nowrap\|1=''n'' = 0}} in this statement, since <math>\mathbb{Z}</math> is isomorphic to <math>\mathbb{Z}/0\mathbb{Z}</math>.{{sfn\|Lang\|2002}}{{rp\|89-90}}

	==Subring test==		== Subring test ==
	The '''subring test''' is a ] that states that for any ring ''R'', a ] ''S'' of ''R'' is a subring if and only if it is ] under multiplication and subtraction, and contains the multiplicative identity of ''R''.{{sfn\|Dummit\|Foote\|2004}}{{rp\|228}}		The '''subring test''' is a ] that states that for any ring ''R'', a ] ''S'' of ''R'' is a subring if and only if it is ] under multiplication and subtraction, and contains the multiplicative identity of ''R''.{{sfn\|Dummit\|Foote\|2004}}{{rp\|228}}

	As an example, the ring '''Z''' of ]s is a subring of the ] of ]s and also a subring of the ring of ]s '''Z'''.		As an example, the ring '''Z''' of ]s is a subring of the ] of ]s and also a subring of the ring of ]s '''Z'''.

	==Ring extensions==		== Ring extensions ==
	{{distinguish\|text=a ring-theoretic analog of a group extension. For that meaning, see an old version of the article ]}}		{{distinguish\|text=a ring-theoretic analog of a group extension. For that meaning, see an old version of the article ]}}

	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 ]s.		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 ]s.

	==Subring generated by a set==		== 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'' ''']''' by ''X''. If ''S'' = ''R,'' we may say that the ring ''R'' is ''generated'' by ''X''.		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'' ''']''' by ''X''. If ''S'' = ''R,'' we may say that the ring ''R'' is ''generated'' by ''X''.

	==Relation to ideals==		== Relation to ideals ==
	Proper ]s are subrings (without unity) that are closed under both left and right multiplication by elements of ''R''.		Proper ]s are subrings (without unity) that are closed under both left and right multiplication by elements of ''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):		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 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.		* The proper ideals of '''Z''' have no multiplicative identity.

	==Profile by commutative subrings==		== Profile by commutative subrings ==
	A ring may be profiled{{clarify\|what "profile" means here?\|date=June 2016}} by the variety of ] subrings that it hosts:		A ring may be profiled{{clarify\|what "profile" means here?\|date=June 2016}} by the variety of ] subrings that it hosts:
	*The ] ring '''H''' contains only the ] as a planar subring		* The ] ring '''H''' contains only the ] as a planar subring
	*The ] ring contains three types of commutative planar subrings: the ] plane, the ] plane, as well as the ordinary complex plane		* The ] ring contains three types of commutative planar subrings: the ] plane, the ] plane, as well as the ordinary complex plane
	*The ] also contains 3-dimensional commutative subrings generated by the ] and a ] ε of order 3 (εεε = 0 ≠ εε). For instance, the ] can be realized as the join of the ] of two of these nilpotent-generated subrings of 3 × 3 matrices.		* The ] also contains 3-dimensional commutative subrings generated by the ] and a ] ε of order 3 (εεε = 0 ≠ εε). For instance, the ] can be realized as the join of the ] of two of these nilpotent-generated subrings of 3 × 3 matrices.

	==See also==		== See also ==
	* ]		* ]
	* ]		* ]
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	* ]		* ]

	==Notes==		== Notes ==
	{{reflist}}		{{reflist}}

	==References==		== References ==
	*{{cite book \|last1=Adamson \|first1=Iain T. \| title=Elementary rings and modules \| series=University Mathematical Texts \| publisher=Oliver and Boyd \| year=1972 \| isbn=0-05-002192-3 \| pages=14–16 }}		* {{cite book \|last1=Adamson \|first1=Iain T. \| title=Elementary rings and modules \| series=University Mathematical Texts \| publisher=Oliver and Boyd \| year=1972 \| isbn=0-05-002192-3 \| pages=14–16 }}
	*{{cite book \|last1=Dummit \|first1=David Steven \|last2=Foote \|first2=Richard Martin \|title=Abstract algebra \|date=2004 \|publisher=John Wiley & Sons \|location=Hoboken, NJ \|isbn=0-471-43334-9 \|edition=Third \|url=https://archive.org/details/abstractalgebra0000dumm_k3c6}}		* {{cite book \|last1=Dummit \|first1=David Steven \|last2=Foote \|first2=Richard Martin \|title=Abstract algebra \|date=2004 \|publisher=John Wiley & Sons \|location=Hoboken, NJ \|isbn=0-471-43334-9 \|edition=Third \|url=https://archive.org/details/abstractalgebra0000dumm_k3c6}}
	*{{cite book \|last1=Lang \|first1=Serge \|title=Algebra \|date=2002 \|location=New York \|isbn=978-0387953854 \|edition=3 \|url=https://archive.org/details/algebra-serge-lang}}		* {{cite book \|last1=Lang \|first1=Serge \|title=Algebra \|date=2002 \|location=New York \|isbn=978-0387953854 \|edition=3 \|url=https://archive.org/details/algebra-serge-lang}}
	*{{cite book \|last1=Sharpe \|first1=David \|title=Rings and factorization \|url=https://archive.org/details/ringsfactorizati0000shar \| url-access=registration \| publisher=] \| year=1987 \| isbn=0-521-33718-6 \| pages=}}		* {{cite book \|last1=Sharpe \|first1=David \|title=Rings and factorization \|url=https://archive.org/details/ringsfactorizati0000shar \| url-access=registration \| publisher=] \| year=1987 \| isbn=0-521-33718-6 \| pages=}}

	]		]