Ore extension

<h2 id="definition">Definition</h2>
Suppose that R is a (not necessarily <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a>) <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>, 
 
 
 
 σ
 :
 R
 →
 R
 
 
 {\displaystyle \sigma \colon R\to R}
 
 is a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a>, and 
 
 
 
 δ
 :
 R
 →
 R
 
 
 {\displaystyle \delta \colon R\to R}
 
 is a σ-derivation of R, which means that 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 is a <a href="/facts/Group_homomorphism/sDguaNYs">homomorphism</a> of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> satisfying

δ
 (
 
 r
 
 1
 
 
 
 r
 
 2
 
 
 )
 =
 σ
 (
 
 r
 
 1
 
 
 )
 δ
 (
 
 r
 
 2
 
 
 )
 +
 δ
 (
 
 r
 
 1
 
 
 )
 
 r
 
 2
 
 
 
 
 {\displaystyle \delta (r_{1}r_{2})=\sigma (r_{1})\delta (r_{2})+\delta (r_{1})r_{2}}
 
.
Then the Ore extension 
 
 
 
 R
 [
 x
 ;
 σ
 ,
 δ
 ]
 
 
 {\displaystyle R[x;\sigma ,\delta ]}
 
, also called a skew polynomial ring, is the <a href="/facts/Noncommutative_ring/3jmll7q7">noncommutative ring</a> obtained by giving the <a href="/facts/Polynomial_ring/ua3vk8Ih">ring of polynomials</a> 
 
 
 
 R
 [
 x
 ]
 
 
 {\displaystyle R[x]}
 
 a new multiplication, subject to the identity

x
 r
 =
 σ
 (
 r
 )
 x
 +
 δ
 (
 r
 )
 
 
 {\displaystyle xr=\sigma (r)x+\delta (r)}
 
.
If δ = 0 (i.e., is the zero map) then the Ore extension is denoted R[x; σ]. If σ = 1 (i.e., the <a href="/facts/Identity_map/9AtsP0LC">identity map</a>) then the Ore extension is denoted R[ x, δ ] and is called a differential polynomial ring.

<h2 id="examples">Examples</h2>
The <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebras</a> are Ore extensions, with R any commutative <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a>, σ the identity ring <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a>, and δ the <a href="/facts/Formal_derivative/actYz19a">polynomial derivative</a>. <a href="/facts/Ore_algebra/gezGI3jk">Ore algebras</a> are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of <a href="/facts/Gr%C3%B6bner_bases/efMrjM7w">Gröbner bases</a>.

<h2 id="properties">Properties</h2>
<ul><li>An Ore extension of a <a href="/facts/Domain_(ring_theory)/p4eKRC8W">domain</a> is a domain.</li>
<li>An Ore extension of a <a href="/facts/Skew_field/bCf07uRj">skew field</a> is a non-commutative <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domain</a>.</li>
<li>If σ is an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> and R is a left <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> then the Ore extension R[ λ; σ, δ ] is also left Noetherian.</li></ul>
<h2 id="elements">Elements</h2>
An element f of an Ore ring R is called

<ul><li>twosided<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> (or invariant<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> ), if R·f = f·R, and</li>
<li>central, if g·f = f·g for all g in R.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Goodearl, K. R.; Warfield, R. B. Jr. (2004), An Introduction to Noncommutative Noetherian Rings, Second Edition, London Mathematical Society Student Texts, vol. 61, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-54537-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2080008">2080008</a></li>
<li>McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian rings, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 30, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-2169-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1811901">1811901</a></li>
<li>Azeddine Ouarit (1992) Extensions de ore d'anneaux noetheriens á i.p, Comm. Algebra, 20 No 6,1819-1837. <a href="https://zbmath.org/?q=an:0754.16014">https://zbmath.org/?q=an:0754.16014</a></li>
<li>Azeddine Ouarit (1994) A remark on the Jacobson property of PI Ore extensions. (Une remarque sur la propriété de Jacobson des extensions de Ore a I.P.) (French) Zbl 0819.16024. Arch. Math. 63, No.2, 136-139 (1994). <a href="https://zbmath.org/?q=an:00687054">https://zbmath.org/?q=an:00687054</a></li>
<li>Rowen, Louis H. (1988), Ring theory, vol. I, II, Pure and Applied Mathematics, vol. 127, 128, Boston, MA: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-599841-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0940245">0940245</a></li></ul>

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

<ol>
<li id="fn:1">Jacobson, Nathan (1996). Finite-Dimensional Division Algebras over Fields. Springer. <a href="/wiki/Nathan_Jacobson" target="_blank">/wiki/Nathan_Jacobson</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cohn, Paul M. (1995). Skew Fields: Theory of General Division Rings. Cambridge University Press. <a href="/wiki/Paul_Cohn" target="_blank">/wiki/Paul_Cohn</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Ore extension open-in-new

Ore extension