Subring

In mathematics, a '''subring''' of a ring {{mvar|R}} is a subset of {{mvar|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 {{mvar|R}}.<ref group=lower-alpha>In general, not all subsets of a ring {{mvar|R}} are rings.</ref>

== Definition == A subring of a ring {{math|(''R'', +, *, 0, 1)}} is a subset {{mvar|S}} of {{mvar|R}} that preserves the structure of the ring, i.e. a ring {{math|(''S'', +, *, 0, 1)}} with {{math|''S'' ⊆ ''R''}}. Equivalently, it is both a subgroup of {{math|(''R'', +, 0)}} and a submonoid of {{math|(''R'', *, 1)}}.

Equivalently, {{mvar|S}} is a subring if and only if it contains the multiplicative identity of {{mvar|R}}, and is closed under multiplication and subtraction. This is sometimes known as the ''subring test''.<ref name="Dummit & Foote">{{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 |page=228}}</ref>

=== Variations === Some mathematicians define rings without requiring the existence of a multiplicative identity (see ''{{slink|Ring (mathematics)|History}}''). In this case, a subring of {{mvar|R}} is a subset of {{mvar|R}} that is a ring for the operations of {{mvar|R}} (this does imply it contains the additive identity of {{mvar|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 {{mvar|R}}. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of {{mvar|R}} that is a subring of {{mvar|R}} is {{mvar|R}} itself.

== Examples == * The ring of integers <math>\Z</math> is a subring of both the field of real numbers and the polynomial ring <math>\Z[X]</math>.<ref name="Dummit & Foote" />

* <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.<ref name="Dummit & Foote" />

* Every ring has a unique smallest subring, isomorphic to some ring <math>\mathbb{Z}/n\mathbb{Z}</math> with ''n'' a nonnegative integer (see ''Characteristic''). 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>.<ref>{{cite book |last1=Lang |first1=Serge |title=Algebra |date=2002 |location=New York |isbn=978-0387953854 |edition=3 |pages=89–90}}</ref>

* The center of a ring {{mvar|R}} is a subring of {{mvar|R}}, and {{mvar|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 {{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 name="lovett">{{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 linear combinations with integer coefficients of products of elements of {{mvar|X}}, including the additive identity ("empty combination") and multiplicative identity ("empty product").<ref>{{cite book|title=Abstract Algebra: An Introduction with Applications|first=Derek J. S.|last=Robinson|edition=3rd|publisher=Walter de Gruyter GmbH & Co KG|year=2022|isbn=9783110691160|page=109|url=https://books.google.com/books?id=DOxcEAAAQBAJ&pg=PA109}}</ref>

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''}}.

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}}.

== Ring extension == Subrings generalize some aspects of field extensions. If {{mvar|S}} is a subring of a ring {{mvar|R}}, then equivalently {{mvar|R}} is said to be a '''ring extension'''<ref group=lower-alpha>Not to be confused with the ring-theoretic analog of a group extension.</ref> of {{mvar|S}}.

=== Adjoining === If {{mvar|A}} is a ring and {{mvar|T}} is a subring of {{mvar|A}} generated by {{math|''R'' ∪ ''S''}}, where {{mvar|R}} is a subring, then {{mvar|T}} is a ring extension and is said to be {{mvar|S}} ''adjoined to'' {{mvar|R}}, denoted {{math|''R''[''S'']}}. Individual elements can also be adjoined to a subring, denoted {{math|''R''[''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''n''}}]}}.<ref>{{cite book |last=Gouvêa |first=Fernando Q. |author-link=Fernando Q. Gouvêa |date=2012 |title=A Guide to Groups, Rings, and Fields |chapter=Rings and Modules |page=145 |publisher=Mathematical Association of America |publication-place=Washington, DC |isbn=9780883853559}}</ref><ref name="lovett" />

For example, the ring of Gaussian integers <math>\Z[i]</math> is a subring of <math>\C</math> generated by <math>\Z \cup \{i\}</math>, and thus is the adjunction of the imaginary unit {{mvar|i}} to <math>\Z</math>.<ref name="lovett" />

=== Prime subring === The intersection of all subrings of a ring {{mvar|R}} is a subring that may be called the ''prime subring'' of {{mvar|R}} by analogy with prime fields.

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

== Notes == {{notelist-la}}

== References == {{reflist}}

=== General references === * {{cite book |last1=Adamson |first1=Iain T. |title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd |date=1972 |isbn=0-05-002192-3 |pages=14–16}} * {{cite book |last1=Sharpe |first1=David |title=Rings and factorization |url=https://archive.org/details/ringsfactorizati0000shar | url-access=registration | publisher=Cambridge University Press |date=1987 |isbn=0-521-33718-6 | pages=[https://archive.org/details/ringsfactorizati0000shar/page/15 15–17]}}

Category:Ring theory