Subring

<h2 id="definition">Definition</h2>
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 <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of (R, +, 0) and a <a href="/facts/Submonoid/6AHve7p8">submonoid</a> of (R, *, 1).
Equivalently, S is a subring <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it contains the multiplicative identity of R, and is <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under multiplication and subtraction. This is sometimes known as the subring test.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h3>Variations</h3>
Some mathematicians define rings without requiring the existence of a multiplicative identity (see <a href="/facts/Ring_(mathematics)/JnCUXz63">Ring (mathematics) § History</a>). 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 <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a> 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.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 is a subring of both the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> 
 
 
 
 
 Z
 
 [
 X
 ]
 
 
 {\displaystyle \mathbb {Z} [X]}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
<ul><li>
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 and its quotients 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
 have no subrings (with multiplicative identity) other than the full ring.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li></ul>
<ul><li>Every ring has a unique smallest subring, isomorphic to some ring 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
 with n a nonnegative integer (see <a href="/facts/Characteristic_(algebra)/D2VzcQaG">Characteristic</a>). The integers 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 correspond to n = 0 in this statement, since 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 is isomorphic to 
 
 
 
 
 Z
 
 
 /
 
 0
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /0\mathbb {Z} }
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<ul><li>The <a href="/facts/Center_of_a_ring/kJcejgkj">center of a ring</a> R is a subring of R, and R is an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over its center.</li></ul>
<h2 id="subring-generated-by-a-set">Subring generated by a set</h2>
See also: <a href="/facts/Generator_(mathematics)/Jo1eBRbD">Generator (mathematics)</a>
A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The subring generated by X is also the set of all <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> with integer coefficients of products of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Any intersection of subrings of R is itself a subring of R; therefore, the subring generated by X (denoted here as S) is indeed a subring of R. This subring S is the smallest subring of R containing X; that is, if T is any other subring of R containing X, then S ⊆ T.
Since R itself is a subring of R, if R is generated by X, it is said that the ring R is generated by X.

<h2 id="ring-extension">Ring extension</h2>
Subrings generalize some aspects of <a href="/facts/Field_extension/L7yIDtyK">field extensions</a>. If S is a subring of a ring R, then equivalently R is said to be a ring extension<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> of S.

<h3>Adjoining</h3>
If A is a ring and T is a subring of A generated by R ∪ S, where R is a subring, then T is a ring extension and is said to be S adjoined to R, denoted R[S]. Individual elements can also be adjoined to a subring, denoted R[a1, a2, ..., an].<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
For example, the ring of <a href="/facts/Gaussian_integers/zj82AhtL">Gaussian integers</a> 
 
 
 
 
 Z
 
 [
 i
 ]
 
 
 {\displaystyle \mathbb {Z} [i]}
 
 is a subring of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 generated by 
 
 
 
 
 Z
 
 ∪
 {
 i
 }
 
 
 {\displaystyle \mathbb {Z} \cup \{i\}}
 
, and thus is the adjunction of the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> i to 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h3>Prime subring</h3>
The intersection of all subrings of a ring R is a subring that may be called the prime subring of R by analogy with <a href="/facts/Prime_field/D2VzcQaG">prime fields</a>. 
The prime subring of a ring R is a subring of the center of R, which is <a href="/facts/Ring_isomorphism/UptcMpFk">isomorphic</a> either to the ring 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 of the <a href="/facts/Integers/PUwwrawx">integers</a> or to the ring of the <a href="/facts/Modular_arithmetic/0wtRa5dU">integers modulo n</a>, where n is the smallest positive integer such that the sum of n copies of 1 equals 0.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Integral_extension/55dAuE39">Integral extension</a></li>
<li><a href="/facts/Group_extension/pGoe0l4d">Group extension</a></li>
<li><a href="/facts/Algebraic_extension/GAs76USY">Algebraic extension</a></li>
<li><a href="/facts/Ore_extension/CLfG1na6">Ore extension</a></li></ul>
<h2 id="notes">Notes</h2>

<h3>General references</h3>
<ul><li>Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-05-002192-3.</li>
<li>Sharpe, David (1987). <a href="https://archive.org/details/ringsfactorizati0000shar">Rings and factorization</a>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. pp. <a href="https://archive.org/details/ringsfactorizati0000shar/page/15">15–17</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-33718-6.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">In general, not all subsets of a ring R are rings. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9. <a href="0-471-43334-9" target="_blank">0-471-43334-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9. <a href="0-471-43334-9" target="_blank">0-471-43334-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9. <a href="0-471-43334-9" target="_blank">0-471-43334-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-0387953854" target="_blank">978-0387953854</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913. <a href="9781482248913" target="_blank">9781482248913</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Robinson, Derek J. S. (2022). Abstract Algebra: An Introduction with Applications (3rd ed.). Walter de Gruyter GmbH & Co KG. p. 109. ISBN 9783110691160. <a href="9783110691160" target="_blank">9783110691160</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Not to be confused with the ring-theoretic analog of a group extension. <a href="/wiki/Group_extension" target="_blank">/wiki/Group_extension</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Gouvêa, Fernando Q. (2012). "Rings and Modules". A Guide to Groups, Rings, and Fields. Washington, DC: Mathematical Association of America. p. 145. ISBN 9780883853559. <a href="9780883853559" target="_blank">9780883853559</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913. <a href="9781482248913" target="_blank">9781482248913</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913. <a href="9781482248913" target="_blank">9781482248913</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Subring open-in-new

Subring