Unique factorization domain

<h2 id="definition">Definition</h2>
Formally, a unique factorization domain is defined to be an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> R in which every non-zero element x of R which is not a unit can be written as a finite product of <a href="/facts/Irreducible_element/6oMIppy4">irreducible elements</a> pi of R:

x = p1 p2 ⋅⋅⋅ pn with n ≥ 1
and this representation is unique in the following sense:
If q1, ..., qm are irreducible elements of R such that

x = q1 q2 ⋅⋅⋅ qm with m ≥ 1,
then m = n, and there exists a <a href="/facts/Bijective/j4gTuTmW">bijective map</a> φ : {1, ..., n} → {1, ..., m} such that pi is <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">associated</a> to qφ(i) for i ∈ {1, ..., n}.

<h2 id="examples">Examples</h2>
Most rings familiar from elementary mathematics are UFDs:

<ul><li>All <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a>, hence all <a href="/facts/Euclidean_domain/ydwCD0fz">Euclidean domains</a>, are UFDs. In particular, the <a href="/facts/Integers/PUwwrawx">integers</a> (also see <a href="/facts/Fundamental_theorem_of_arithmetic/EzHUwlXX">Fundamental theorem of arithmetic</a>), the <a href="/facts/Gaussian_integer/zj82AhtL">Gaussian integers</a> and the <a href="/facts/Eisenstein_integer/UGQmFI8c">Eisenstein integers</a> are UFDs.</li>
<li>If R is a UFD, then so is R[X], the <a href="/facts/Polynomial_ring/ua3vk8Ih">ring of polynomials</a> with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.</li>
<li>The <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> ring K[[X1, ..., Xn]] over a field K (or more generally over a <a href="/facts/Regular_local_ring/P9s5Fvrl">regular</a> UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is <a href="/facts/Local_ring/gRTdg5Qx">local</a>. For example, if R is the localization of k[x, y, z]/(x2 + y3 + z7) at the <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> (x, y, z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD.</li>
<li>The <a href="/facts/Auslander%25E2%2580%2593Buchsbaum_theorem/ym74CQRv">Auslander–Buchsbaum theorem</a> states that every <a href="/facts/Regular_local_ring/P9s5Fvrl">regular local ring</a> is a UFD.</li>
<li>
 
 
 
 
 Z
 
 
 [
 
 e
 
 
 
 2
 π
 i
 
 n
 
 
 
 ]
 
 
 
 {\displaystyle \mathbb {Z} \left[e^{\frac {2\pi i}{n}}\right]}
 
 is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.</li>
<li>Mori showed that if the completion of a <a href="/facts/Zariski_ring/nSqun41w">Zariski ring</a>, such as a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian local ring</a>, is a UFD, then the ring is a UFD.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a> of k[x, y, z]/(x2 + y3 + z5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x2 + y3 + z7) at the prime ideal (x, y, z) the local ring is a UFD but its completion is not.</li>
<li>Let 
 
 
 
 R
 
 
 {\displaystyle R}
 
 be a field of any characteristic other than 2. Klein and Nagata showed that the ring R[X1, ..., Xn]/Q is a UFD whenever Q is a nonsingular quadratic form in the Xs and n is at least 5. When n = 4, the ring need not be a UFD. For example, R[X, Y, Z, W]/(XY − ZW) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles.</li>
<li>The ring Q[x, y]/(x2 + 2y2 + 1) is a UFD, but the ring Q(i)[x, y]/(x2 + 2y2 + 1) is not. On the other hand, The ring Q[x, y]/(x2 + y2 − 1) is not a UFD, but the ring Q(i)[x, y]/(x2 + y2 − 1) is.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Similarly the <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> R[X, Y, Z]/(X2 + Y2 + Z2 − 1) of the 2-dimensional <a href="/facts/Sphere/4Ivb1cTj">real sphere</a> is a UFD, but the coordinate ring C[X, Y, Z]/(X2 + Y2 + Z2 − 1) of the complex sphere is not.</li>
<li>Suppose that the variables Xi are given weights wi, and F(X1, ..., Xn) is a <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomial</a> of weight w. Then if c is coprime to w and R is a UFD and either every <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> <a href="/facts/Projective_module/lHymqhNC">projective module</a> over R is <a href="/facts/Free_module/o6rC6yoJ">free</a> or c is 1 mod w, the ring R[X1, ..., Xn, Z]/(Zc − F(X1, ..., Xn)) is a UFD.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
<h3>Non-examples</h3>
<ul><li>The <a href="/facts/Quadratic_integer_ring/uGsLh5kL">quadratic integer ring</a> 
 
 
 
 
 Z
 
 [
 
 
 −
 5
 
 
 ]
 
 
 {\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
 
 of all <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> of the form 
 
 
 
 a
 +
 b
 
 
 −
 5
 
 
 
 
 {\displaystyle a+b{\sqrt {-5}}}
 
, where a and b are integers, is not a UFD because 6 factors as both 2×3 and as 
 
 
 
 
 (
 
 1
 +
 
 
 −
 5
 
 
 
 )
 
 
 (
 
 1
 −
 
 
 −
 5
 
 
 
 )
 
 
 
 {\displaystyle \left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right)}
 
. These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, 
 
 
 
 1
 +
 
 
 −
 5
 
 
 
 
 {\displaystyle 1+{\sqrt {-5}}}
 
, and 
 
 
 
 1
 −
 
 
 −
 5
 
 
 
 
 {\displaystyle 1-{\sqrt {-5}}}
 
 are <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">associate</a>. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> See also <a href="/facts/Algebraic_integer/Qb0fD3R3">Algebraic integer</a>.</li>
<li>For a <a href="/facts/Square-free_integer/R3qR9hP1">square-free positive integer</a> d, the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of 
 
 
 
 
 Q
 
 [
 
 
 −
 d
 
 
 ]
 
 
 {\displaystyle \mathbb {Q} [{\sqrt {-d}}]}
 
 will fail to be a UFD unless d is a <a href="/facts/Heegner_number/ExPvl9UM">Heegner number</a>.</li>
<li>The ring of formal power series over the complex numbers is a UFD, but the <a href="/facts/Subring/M0perybt">subring</a> of those that converge everywhere, in other words the ring of <a href="/facts/Entire_function/XNluKzu7">entire functions</a> in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.:

sin
 ⁡
 π
 z
 =
 π
 z
 
 ∏
 
 n
 =
 1
 
 
 ∞
 
 
 
 (
 
 1
 −
 
 
 
 
 z
 
 2
 
 
 
 
 
 n
 
 2
 
 
 
 
 
 
 )
 
 .
 
 
 {\displaystyle \sin \pi z=\pi z\prod _{n=1}^{\infty }\left(1-{{z^{2}} \over {n^{2}}}\right).}
 
</li></ul>
<h2 id="properties">Properties</h2>
Some concepts defined for integers can be generalized to UFDs:

<ul><li>In UFDs, every <a href="/facts/Irreducible_element/6oMIppy4">irreducible element</a> is <a href="/facts/Prime_element/RTuk5fjg">prime</a>. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈ K[x, y, z]/(z2 − xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the <a href="/facts/Ascending_chain_condition_on_principal_ideals/7MSUGgWp">ACCP</a> is a UFD if and only if every irreducible element is prime.</li>
<li>Any two elements of a UFD have a <a href="/facts/Greatest_common_divisor/kNFvBuKl">greatest common divisor</a> and a <a href="/facts/Least_common_multiple/AX1jIUnw">least common multiple</a>. Here, a greatest common divisor of a and b is an element d that <a href="/facts/Divisor/DKNa2GvC">divides</a> both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are <a href="/facts/Associated_element/cylt6zxW">associated</a>.</li>
<li>Any UFD is <a href="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed</a>. In other words, if R is a UFD with <a href="/facts/Quotient_field/cmlJrDSS">quotient field</a> K, and if an element k in K is a <a href="/facts/Polynomial_root/mlK3dhoT">root</a> of a <a href="/facts/Monic_polynomial/4JzuIJ5R">monic polynomial</a> with <a href="/facts/Coefficients/5Hwq2Wuq">coefficients</a> in R, then k is an element of R.</li>
<li>Let S be a <a href="/facts/Multiplicatively_closed_subset/WH1VI3b3">multiplicatively closed subset</a> of a UFD A. Then the <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a> S−1A is a UFD. A partial converse to this also holds; see below.</li></ul>
<h2 id="equivalent-conditions-for-a-ring-to-be-a-ufd">Equivalent conditions for a ring to be a UFD</h2>
A <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> integral domain is a UFD if and only if every <a href="/facts/Height_(ring_theory)/KQb2uLtz">height</a> 1 <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> is principal (a proof is given at the end). Also, a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a> is a UFD if and only if its <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a> is trivial. In this case, it is in fact a <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domain</a>.
In general, for an integral domain A, the following conditions are equivalent:

<ol><li>A is a UFD.</li>
<li>Every nonzero <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> of A contains a <a href="/facts/Prime_element/RTuk5fjg">prime element</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>A satisfies <a href="/facts/Ascending_chain_condition_on_principal_ideals/7MSUGgWp">ascending chain condition on principal ideals</a> (ACCP), and the <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a> S−1A is a UFD, where S is a <a href="/facts/Multiplicatively_closed_subset/WH1VI3b3">multiplicatively closed subset</a> of A generated by prime elements. (Nagata criterion)</li>
<li>A satisfies <a href="/facts/Ascending_chain_condition_on_principal_ideals/7MSUGgWp">ACCP</a> and every <a href="/facts/Irreducible_element/6oMIppy4">irreducible</a> is <a href="/facts/Prime_element/RTuk5fjg">prime</a>.</li>
<li>A is <a href="/facts/Atomic_domain/VX8jcsAI">atomic</a> and every <a href="/facts/Irreducible_element/6oMIppy4">irreducible</a> is <a href="/facts/Prime_element/RTuk5fjg">prime</a>.</li>
<li>A is a <a href="/facts/GCD_domain/PWrf9Tkm">GCD domain</a> satisfying <a href="/facts/Ascending_chain_condition_on_principal_ideals/7MSUGgWp">ACCP</a>.</li>
<li>A is a <a href="/facts/Schreier_domain/po9LYu2l">Schreier domain</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and <a href="/facts/Atomic_domain/VX8jcsAI">atomic</a>.</li>
<li>A is a <a href="/facts/Schreier_domain/po9LYu2l">pre-Schreier domain</a> and <a href="/facts/Atomic_domain/VX8jcsAI">atomic</a>.</li>
<li>A has a divisor theory in which every divisor is principal.</li>
<li>A is a <a href="/facts/Krull_domain/fF9TWfaM">Krull domain</a> in which every <a href="/facts/Divisorial_ideal/I5rvJ6nt">divisorial ideal</a> is principal (in fact, this is the definition of UFD in Bourbaki.)</li>
<li>A is a Krull domain and every prime ideal of height 1 is principal.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ol>
In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID.
For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains a height one prime ideal (induction on height) that is principal. By (2), the ring is a UFD.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Parafactorial_local_ring/wWpVX7k8">Parafactorial local ring</a></li>
<li><a href="/facts/Noncommutative_unique_factorization_domain/INWCqrhg">Noncommutative unique factorization domain</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (2011). Algebra. Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-241377-0.</li>
<li>Bourbaki, N. (1972). <a href="https://archive.org/details/commutativealgeb0000bour">Commutative algebra</a>. Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780201006445.</li>
<li><a href="/facts/Harold_Edwards_(mathematician)/JKfNgfJB">Edwards, Harold M.</a> (1990). Divisor Theory. Boston: Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3448-3.</li>
<li><a href="/facts/Brian_Hartley/W6oE3L02">Hartley, B.</a>; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-09810-5. Chap. 4.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-55540-0, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0848.13001">0848.13001</a> Chapter II.5</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>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-33718-6.</li>
<li><a href="/facts/Pierre_Samuel/r63UjoTx">Samuel, Pierre</a> (1964), Murthy, M. Pavman (ed.), <a href="http://www.math.tifr.res.in/~publ/ln/">Lectures on unique factorization domains</a>, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0214579">0214579</a></li>
<li><a href="/facts/Pierre_Samuel/r63UjoTx">Samuel, Pierre</a> (1968). "Unique factorization". <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">The American Mathematical Monthly</a>. 75 (9): 945–952. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2315529">10.2307/2315529</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2315529">2315529</a>.</li>
<li>Weintraub, Steven H. (2008). Factorization: Unique and Otherwise. Wellesley, Mass.: A K Peters/CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-241-0.</li></ul>

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

<ol>
<li id="fn:1">Bourbaki (1972), 7.3, no 6, Proposition 4 - Bourbaki, N. (1972). Commutative algebra. Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co. ISBN 9780201006445. <a href="https://archive.org/details/commutativealgeb0000bour" target="_blank">https://archive.org/details/commutativealgeb0000bour</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Samuel (1964), p. 35 - Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, MR 0214579 <a href="http://www.math.tifr.res.in/~publ/ln/" target="_blank">http://www.math.tifr.res.in/~publ/ln/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Samuel (1964), p. 31 - Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, MR 0214579 <a href="http://www.math.tifr.res.in/~publ/ln/" target="_blank">http://www.math.tifr.res.in/~publ/ln/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Artin (2011), p. 360 - Artin, Michael (2011). Algebra. Prentice Hall. ISBN 978-0-13-241377-0. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Kaplansky <a href="/wiki/Irving_Kaplansky" target="_blank">/wiki/Irving_Kaplansky</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Bourbaki (1972), 7.3, no 2, Theorem 1. - Bourbaki, N. (1972). Commutative algebra. Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co. ISBN 9780201006445. <a href="https://archive.org/details/commutativealgeb0000bour" target="_blank">https://archive.org/details/commutativealgeb0000bour</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Unique factorization domain open-in-new

Unique factorization domain