Subring#Subring generated by a set

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.{{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}}

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.

• The ring of integers $\backslash Z$ is a subring of both the field of real numbers and the polynomial ring $\backslash Z[X]$.

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

• Every ring has a unique smallest subring, isomorphic to some ring $\backslash mathbb\{Z\}/n\backslash mathbb\{Z\}$ with n a nonnegative integer (see Characteristic). The integers $\backslash mathbb\{Z\}$ correspond to {{nowrap|1=n = 0}} in this statement, since $\backslash mathbb\{Z\}$ is isomorphic to $\backslash mathbb\{Z\}/0\backslash mathbb\{Z\}$.{{cite book |last1=Lang |first1=Serge |title=Algebra |date=2002 |location=New York |isbn=978-0387953854 |edition=3 |pages=89–90}}

• The center of a ring {{mvar|R}} is a subring of {{mvar|R}}, and {{mvar|R}} is an associative algebra over its center.

{{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}}.{{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}} 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").{{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}}

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}}.

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 extensionNot to be confused with the ring-theoretic analog of a group extension. of {{mvar|S}}.

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}}]}}.{{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}}

For example, the ring of Gaussian integers $\backslash Z[i]$ is a subring of $\backslash C$ generated by $\backslash Z\; \backslash cup\; \backslash \{i\backslash \}$, and thus is the adjunction of the imaginary unit {{mvar|i}} to $\backslash Z$.

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 $\backslash Z$ of the integers or to the ring of the modular arithmetic, where {{mvar|n}} is the smallest positive integer such that the sum of {{mvar|n}} copies of {{math|1}} equals {{math|0}}.

• Integral extension
• Group extension
• Algebraic extension
• Ore extension

{{notelist-la}}

{{reflist}}

• {{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