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Fractional ideal
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<h2 id="definition-and-basic-results">Definition and basic results</h2> <p>Let R {\displaystyle R} be an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a>, and let K = Frac R {\displaystyle K=\operatorname {Frac} R} be its <a href="/facts/Field_of_fractions/cmlJrDSS">field of fractions</a>. </p><p>A fractional ideal of R {\displaystyle R} is an R {\displaystyle R} -<a href="/facts/Submodule/jP0LIhFL">submodule</a> I {\displaystyle I} of K {\displaystyle K} such that there exists a non-zero r ∈ R {\displaystyle r\in R} such that r I ⊆ R {\displaystyle rI\subseteq R} . The element r {\displaystyle r} can be thought of as clearing out the denominators in I {\displaystyle I} , hence the name fractional ideal. </p><p>The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle K} generated by a single nonzero element of K {\displaystyle K} . A fractional ideal I {\displaystyle I} is contained in R {\displaystyle R} <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is an (integral) ideal of R {\displaystyle R} . </p><p>A fractional ideal I {\displaystyle I} is called invertible if there is another fractional ideal J {\displaystyle J} such that </p> I J = R {\displaystyle IJ=R} <p>where </p> I J = { a 1 b 1 + a 2 b 2 + ⋯ + a n b n : a i ∈ I , b j ∈ J , n ∈ Z > 0 } {\displaystyle IJ=\{a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}:a_{i}\in I,b_{j}\in J,n\in \mathbb {Z} _{>0}\}} <p>is the product of the two fractional ideals. </p><p>In this case, the fractional ideal J {\displaystyle J} is uniquely determined and equal to the generalized <a href="/facts/Ideal_quotient/ed3GzqUb">ideal quotient</a> </p> ( R : K I ) = { x ∈ K : x I ⊆ R } . {\displaystyle (R:_{K}I)=\{x\in K:xI\subseteq R\}.} <p>The set of invertible fractional ideals form an <a href="/facts/Commutative_group/FfWwmx4R">commutative group</a> with respect to the above product, where the identity is the <a href="/facts/Unit_ideal/vCvytduX">unit ideal</a> ( 1 ) = R {\displaystyle (1)=R} itself. This group is called the group of fractional ideals of R {\displaystyle R} . The principal fractional ideals form a <a href="/facts/Subgroup/r91mBgpa">subgroup</a>. A (nonzero) fractional ideal is invertible if and only if it is <a href="/facts/Projective_module/lHymqhNC">projective</a> as an R {\displaystyle R} -<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 <a href="/facts/Vector_bundle_(algebraic_geometry)/h2NoUN7e">vector bundle</a> over the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">affine scheme</a> Spec ( R ) {\displaystyle {\text{Spec}}(R)} . </p><p>Every <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> <i>R</i>-submodule of <i>K</i> is a fractional ideal and if R {\displaystyle R} is <a href="/facts/Noetherian_ring/kYbRSdnp">noetherian</a> these are all the fractional ideals of R {\displaystyle R} . </p> <h2 id="dedekind-domains">Dedekind domains</h2> <p>In <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domains</a>, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains: </p> An integral domain is a Dedekind domain if and only if every non-zero fractional ideal is invertible. <p>The set of fractional ideals over a Dedekind domain R {\displaystyle R} is denoted Div ( R ) {\displaystyle {\text{Div}}(R)} . </p><p>Its <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a>. </p> <h2 id="number-fields">Number fields</h2> <p>For the special case of <a href="/facts/Algebraic_number_field/e0K0TXSh">number fields</a> K {\displaystyle K} (such as Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} , where ζ n {\displaystyle \zeta _{n}} = <i>exp(2π i/n)</i>) there is an associated <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> denoted O K {\displaystyle {\mathcal {O}}_{K}} called the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of K {\displaystyle K} . For example, O Q ( d ) = Z [ d ] {\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d}}\,)}=\mathbb {Z} [{\sqrt {d}}\,]} for d {\displaystyle d} <a href="/facts/Squarefree_integer/R3qR9hP1">square-free</a> and <a href="/facts/Modular_arithmetic/0wtRa5dU">congruent</a> to 2 , 3 ( mod 4 ) {\displaystyle 2,3{\text{ }}({\text{mod }}4)} . The key property of these rings O K {\displaystyle {\mathcal {O}}_{K}} is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a> is the study of such groups of class rings. </p> <h3>Associated structures</h3> <p>For the ring of integers<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>pg 2 O K {\displaystyle {\mathcal {O}}_{K}} of a number field, the group of fractional ideals forms a group denoted I K {\displaystyle {\mathcal {I}}_{K}} and the subgroup of principal fractional ideals is denoted P K {\displaystyle {\mathcal {P}}_{K}} . The <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a> is the group of fractional ideals modulo the principal fractional ideals, so </p> C K := I K / P K {\displaystyle {\mathcal {C}}_{K}:={\mathcal {I}}_{K}/{\mathcal {P}}_{K}} <p>and its class number h K {\displaystyle h_{K}} is the <a href="/facts/Order_of_a_group/Cl3tKGFg">order</a> of the group, h K = | C K | {\displaystyle h_{K}=|{\mathcal {C}}_{K}|} . In some ways, the class number is a measure for how "far" the ring of integers O K {\displaystyle {\mathcal {O}}_{K}} is from being a <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domain</a> (UFD). This is because h K = 1 {\displaystyle h_{K}=1} if and only if O K {\displaystyle {\mathcal {O}}_{K}} is a UFD. </p> <h4>Exact sequence for ideal class groups</h4> <p>There is an <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a> </p> 0 → O K ∗ → K ∗ → I K → C K → 0 {\displaystyle 0\to {\mathcal {O}}_{K}^{*}\to K^{*}\to {\mathcal {I}}_{K}\to {\mathcal {C}}_{K}\to 0} <p>associated to every number field. </p> <h3>Structure theorem for fractional ideals</h3> <p>One of the important structure theorems for fractional ideals of a <a href="/facts/Number_field/e0K0TXSh">number field</a> states that every fractional ideal I {\displaystyle I} decomposes uniquely up to ordering as </p> I = ( p 1 … p n ) ( q 1 … q m ) − 1 {\displaystyle I=({\mathfrak {p}}_{1}\ldots {\mathfrak {p}}_{n})({\mathfrak {q}}_{1}\ldots {\mathfrak {q}}_{m})^{-1}} <p>for <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> </p> p i , q j ∈ Spec ( O K ) {\displaystyle {\mathfrak {p}}_{i},{\mathfrak {q}}_{j}\in {\text{Spec}}({\mathcal {O}}_{K})} . <p>in the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of O K {\displaystyle {\mathcal {O}}_{K}} . For example, </p> 2 5 O Q ( i ) {\displaystyle {\frac {2}{5}}{\mathcal {O}}_{\mathbb {Q} (i)}} factors as ( 1 + i ) ( 1 − i ) ( ( 1 + 2 i ) ( 1 − 2 i ) ) − 1 {\displaystyle (1+i)(1-i)((1+2i)(1-2i))^{-1}} <p>Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some α {\displaystyle \alpha } to get an ideal J {\displaystyle J} . Hence </p> I = 1 α J {\displaystyle I={\frac {1}{\alpha }}J} <p>Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements. We call a fractional ideal which is a subset of O K {\displaystyle {\mathcal {O}}_{K}} <i>integral</i>. </p> <h2 id="examples">Examples</h2> <ul><li> 5 4 Z {\displaystyle {\frac {5}{4}}\mathbb {Z} } is a fractional ideal over Z {\displaystyle \mathbb {Z} } </li> <li>For K = Q ( i ) {\displaystyle K=\mathbb {Q} (i)} the ideal ( 5 ) {\displaystyle (5)} splits in O Q ( i ) = Z [ i ] {\displaystyle {\mathcal {O}}_{\mathbb {Q} (i)}=\mathbb {Z} [i]} as ( 2 − i ) ( 2 + i ) {\displaystyle (2-i)(2+i)} </li> <li>For K = Q ζ 3 {\displaystyle K=\mathbb {Q} _{\zeta _{3}}} we have the factorization ( 3 ) = ( 2 ζ 3 + 1 ) 2 {\displaystyle (3)=(2\zeta _{3}+1)^{2}} . This is because if we multiply it out, we get ( 2 ζ 3 + 1 ) 2 = 4 ζ 3 2 + 4 ζ 3 + 1 = 4 ( ζ 3 2 + ζ 3 ) + 1 {\displaystyle {\begin{aligned}(2\zeta _{3}+1)^{2}&=4\zeta _{3}^{2}+4\zeta _{3}+1\\&=4(\zeta _{3}^{2}+\zeta _{3})+1\end{aligned}}} </li></ul> Since ζ 3 {\displaystyle \zeta _{3}} satisfies ζ 3 2 + ζ 3 = − 1 {\displaystyle \zeta _{3}^{2}+\zeta _{3}=-1} , our factorization makes sense. <ul><li>For K = Q ( − 23 ) {\displaystyle K=\mathbb {Q} ({\sqrt {-23}})} we can multiply the fractional ideals</li></ul> I = ( 2 , 1 2 − 23 − 1 2 ) {\displaystyle I=(2,{\frac {1}{2}}{\sqrt {-23}}-{\frac {1}{2}})} and J = ( 4 , 1 2 − 23 + 3 2 ) {\displaystyle J=(4,{\frac {1}{2}}{\sqrt {-23}}+{\frac {3}{2}})} to get the ideal I J = ( 1 2 − 23 + 3 2 ) . {\displaystyle IJ=({\frac {1}{2}}{\sqrt {-23}}+{\frac {3}{2}}).} <h2 id="divisorial-ideal">Divisorial ideal</h2> <p>Let I ~ {\displaystyle {\tilde {I}}} denote the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of all principal fractional ideals containing a nonzero fractional ideal I {\displaystyle I} . </p><p>Equivalently, </p> I ~ = ( R : ( R : I ) ) , {\displaystyle {\tilde {I}}=(R:(R:I)),} <p>where as above </p> ( R : I ) = { x ∈ K : x I ⊆ R } . {\displaystyle (R:I)=\{x\in K:xI\subseteq R\}.} <p>If I ~ = I {\displaystyle {\tilde {I}}=I} then <i>I</i> is called divisorial.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals. </p><p>If <i>I</i> is divisorial and <i>J</i> is a nonzero fractional ideal, then (<i>I</i> : <i>J</i>) is divisorial. </p><p>Let <i>R</i> be a <a href="/facts/Local_ring/gRTdg5Qx">local</a> <a href="/facts/Krull_domain/fF9TWfaM">Krull domain</a> (e.g., a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed</a> local domain). Then <i>R</i> is a <a href="/facts/Discrete_valuation_ring/DaHoTBSH">discrete valuation ring</a> if and only if the <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a> of <i>R</i> is divisorial.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>An integral domain that satisfies the <a href="/facts/Ascending_chain_condition/0K7K2u02">ascending chain conditions</a> on divisorial ideals is called a <a href="/facts/Mori_domain/Gd73efhp">Mori domain</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Divisorial_sheaf/V0QDypeo">Divisorial sheaf</a></li> <li><a href="/facts/Dedekind-Kummer_theorem/upkv7Bw9">Dedekind-Kummer theorem</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Barucci, Valentina (2000), <a href="https://books.google.com/books?id=0tuZkZE07TEC">"Mori domains"</a>, in <a href="/facts/Sarah_Glaz/GGH5awld">Glaz, Sarah</a>; Chapman, Scott T. (eds.), <i>Non-Noetherian commutative ring theory</i>, Mathematics and its Applications, vol. 520, Dordrecht: Kluwer Acad. Publ., pp. 57–73, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7923-6492-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1858157">1858157</a></li> <li>Stein, William, <a href="https://wstein.org/books/ant/ant.pdf"><i>A Computational Introduction to Algebraic Number Theory</i></a> (PDF)</li> <li>Chapter 9 of <a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; <a href="/facts/Ian_G._Macdonald/PoaeNbeg">Macdonald, I.G.</a> (1994), <i>Introduction to Commutative Algebra</i>, Westview Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-40751-8</li> <li>Chapter VII.1 of <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1998), <i>Commutative algebra</i> (2nd ed.), <a href="/facts/Springer_Verlag/nAesf6nT">Springer Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64239-0</li> <li>Chapter 11 of Matsumura, Hideyuki (1989), <i>Commutative ring theory</i>, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-36764-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1011461">1011461</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Childress, Nancy (2009). Class field theory. New York: Springer. ISBN 978-0-387-72490-4. OCLC 310352143. <a href="978-0-387-72490-4" target="_blank">978-0-387-72490-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bourbaki 1998, §VII.1 - Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bourbaki 1998, Ch. VII, § 1, n. 7. Proposition 11. - Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Barucci 2000. - Barucci, Valentina (2000), "Mori domains", in Glaz, Sarah; Chapman, Scott T. (eds.), Non-Noetherian commutative ring theory, Mathematics and its Applications, vol. 520, Dordrecht: Kluwer Acad. Publ., pp. 57–73, ISBN 978-0-7923-6492-4, MR 1858157 <a href="https://books.google.com/books?id=0tuZkZE07TEC" target="_blank">https://books.google.com/books?id=0tuZkZE07TEC</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>