Irreducible element

In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.

The irreducible elements are the terminal elements of a factorization process; that is, they are the factors that cannot be further factorized. If the irreducible factors of every non-zero non-unit element are uniquely defined, up to the multiplication by a unit, then the integral domain is called a unique factorization domain, but this does not need to happen in general for every integral domain. It was discovered in the 19th century that the rings of integers of some number fields are not unique factorization domains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of the same element. The ignorance of this fact is the main error in many of the wrong proofs of Fermat's Last Theorem that were given during the three centuries between Fermat's statement and Wiles's proof of Fermat's Last Theorem.

If $R$ is an integral domain, then $a$ is an irreducible element of $R$ if and only if, for all $b,c\in R$ , the equation $a=bc$ implies that the ideal generated by $a$ is equal to the ideal generated by $b$ or equal to the ideal generated by $c$ . This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no nonzero zero divisors is commonly made in the definition of irreducible elements. It results also that there are several ways to extend the definition of an irreducible element to an arbitrary commutative ring.^[1]

Relationship with prime elements

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element $a$ in a commutative ring $R$ is called prime if, whenever $a\mid bc$ for some $b$ and $c$ in $R,$ then $a\mid b$ or $a\mid c.$ ) In an integral domain, every prime element is irreducible,^[a]^[2] but the converse is not true in general. The converse is true for unique factorization domains^[2] (or, more generally, GCD domains).

Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if $D$ is a GCD domain and $x$ is an irreducible element of $D$ , then as noted above $x$ is prime, and so the ideal generated by $x$ is a prime (hence irreducible) ideal of $D$ .

Example

In the quadratic integer ring $\mathbf {Z} [{\sqrt {-5}}],$ it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,

3\mid \left(2+{\sqrt {-5}}\right)\left(2-{\sqrt {-5}}\right)=9,

but 3 does not divide either of the two factors.^[3]

Notes

^ Consider $p$ a prime element of $R$ and suppose $p=ab.$ Then $p\mid ab,$ so $p\mid a$ or $p\mid b.$ Say $p\mid a,$ so $a=pc$ for some $c\in R$ . Then we have $p=ab=pcb,$ and so $p(1-cb)=0.$ Because $R$ is an integral domain we have $cb=1.$ Therefore $b$ is a unit and $p$ is irreducible.

References

^ Anderson, D. D.; Valdes-Leon, Silvia (1996-06-01). "Factorization in Commutative Rings with Zero Divisors". Rocky Mountain Journal of Mathematics. 26 (2): 439–480. doi:10.1216/rmjm/1181072068. ISSN 0035-7596.
^ ^a ^b Sharpe, David (1987). Rings and factorization. Cambridge University Press. p. 54. ISBN 0-521-33718-6. Zbl 0674.13008.
^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

Read other articles:

Sejarah Māori

Sejarah Māori bermula dengan kedatangan penduduk Polinesia di Selandia Baru (Aotearoa dalam bahasa Māori). Migrasi lintas-samudera ini dilakukan menggunakan kano di antara akhir abad ke-13 hingga awal abad ke-14. Setelah beberapa abad terisolasi, pemukim Polinesia ini kemudian membentuk budaya mereka sendiri. Mereka kemudian dikenal sebagai suku Māori. Sejarah awal Māori umumnya dibagi menjadi dua periode: periode Arkais (sekitar 1300-1500) dan periode Klasik (sekitar 1500-1642). Situs ar...

Jakarta FC 1928

Jakarta FC 1928Nama lengkapJakarta Football Club 1928JulukanMacan MetroBerdiri2010StadionStadion Lebak Bulus, Jakarta(Kapasitas: 25.000)Ketua Umum Hadi BasalamahSekretaris SetiawanManajer Andi TjahjokoPelatih Bambang NurdiansyahLigaLiga Primer Indonesia2011-12Peringkat 9 Kostum kandang Kostum tandang Jakarta Football Club 1928 atau Jakarta FC 1928 adalah sebuah tim sepak bola Indonesia yang berbasis di Jakarta. Klub Ini didirikan tahun 2010 dan bermain di Liga Primer Indonesia 2011. Saat itu,...

Olympus Mons

Olympus MonsKoordinat18°39′N 226°12′E / 18.650°N 226.200°E / 18.650; 226.200Koordinat: 18°39′N 226°12′E / 18.650°N 226.200°E / 18.650; 226.200[1]Puncak219 km (136 mi) ketinggian datum[2] 26 km (16 mi) relief lokal di atas daratan[3]PenemuMariner 9EponimLatin – Mount Olympus Citra Olympus Mons dari wahana antariksa Viking 1 dengan kaldera puncak, lereng curam, dan aureole Gunung Olimpu...

الجامعة الألمانية بالقاهرة

الجامعة الألمانية بالقاهرة شعار الجامعة الألمانية بالقاهرة الشعار (بالإنجليزية: Education for Global Excellence)‏[1] معلومات التأسيس 2002 النوع خاص الكليات الهندسة - الفنون التطبيقية - الصيدلة - الادارة الحقوق و الدراسات القانونية الموقع الجغرافي إحداثيات 29°59′17″N 31°26′31″E...

Kitin

Struktur molekul Kitin Kitin adalah polisakarida struktural yang digunakan untuk menyusun eksoskleton dari artropoda (serangga, laba-laba, krustasea, dan hewan-hewan lain sejenis).[1] Kitin tergolong homopolisakarida linear yang tersusun atas residu N-asetilglukosamin pada rantai beta dan memiliki monomer berupa molekul glukosa dengan cabang yang mengandung nitrogen. [2] Kitin murni mirip dengan kulit, tetapi akan mengeras ketika dilapisi dengan garam kalsium karbonat.[1&#...

Человек прямоходящий

† Человек прямоходящий Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:Синапсиды�...

Lulu White

Brothel madam in New Orleans, Louisiana, US Lulu WhiteNew Orleans Police Department mug shot of Lulu White, 1920Bornca. 1868[1]Alabama, United States[2][1]DiedAugust 20, 1931Other namesLulu HendleyOccupation(s)Brothel owner, prostitute Lulu White (Lulu Hendley, ca. 1868 – August 20, 1931) was a brothel madam, procuress and entrepreneur in New Orleans, Louisiana during the Storyville period.[3] An eccentric figure, she was noted for her love of jewel...

羅德島州

本條目存在以下問題，請協助改善本條目或在討論頁針對議題發表看法。此條目需要擴充。 (2013年1月1日)请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。此條目需要补充更多来源。 (2013年1月1日)请协助補充多方面可靠来源以改善这篇条目，无法查证的内容可能會因為异议提出而被移除。致使用者：请搜索一下条目的...

ヨハネス12世 (ローマ教皇)

ヨハネス12世第130代ローマ教皇教皇就任 955年12月16日教皇離任 964年5月14日先代アガペトゥス2世次代レオ8世個人情報出生 937年スポレート公国（中部イタリア）スポレート死去 964年5月14日教皇領、ローマ原国籍スポレート公国親父アルベリーコ2世（スポレート公）、母アルダその他のヨハネステンプレートを表示ヨハネス12世（Ioannes XII、937年 - 964年5月14日）は、ロ...

1Q84

Artikel ini bukan mengenai Nineteen Eighty-Four. 1Q84 Sampul bagian pertamaPengarangHaruki MurakamiPenerjemahJay RubinPhilip GabrielNegara JepangBahasaJepangGenreSejarah alternatifPenerbitShinchoshaTanggal terbit29 Mei, 2009 (Buku 1 dan 2)16 April, 2010 (Buku 3)Tgl. terbit (bhs. Inggris)25 Oktober, 2011Jenis mediaPrint (sampul keras)Halaman928ISBNISBN 978-0-307-59331-3OCLC701017688 1Q84 edisi Amerika Serikat, pertama kali diterbitkan di Amerika Serikat pada tahun...

Lake Wānaka

Lake in Otago, New Zealand Lake WānakaView of Lake Wānaka from Mt. Roy.Lake WānakaLocation on the South IslandLocationQueenstown-Lakes District, Otago, South IslandCoordinates44°30′S 169°08′E / 44.500°S 169.133°E / -44.500; 169.133Primary inflowsMakarora River, Matukituki RiverPrimary outflowsClutha River / Mata-AuBasin countriesNew ZealandMax. length42 km (26 mi)Max. width10 km (6.2 mi)Surface area192 km2 (74 sq mi)Av...

Edmond Malinvaud

French economist This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Edmond Malinvaud – news · newspapers · books · scholar · JSTOR (March 2015) (Learn how and when to remove this message) Edmond MalinvaudEdmond Malinvaud (right), next to Yves Balasko (center) and Gérard Debreu, in 1977Born(1923-04-25)25 April ...

Compiègne Wagon

Train carriage Ferdinand Foch, second from right, pictured outside the carriage in Compiègne after agreeing to the armistice that ended World War I. The carriage was later chosen by Nazi Germany as the symbolic setting of Pétain's June 1940 armistice.[1] Left to right: Joachim von Ribbentrop, Wilhelm Keitel, Hermann Göring, Rudolf Hess, Adolf Hitler, Erich Raeder partially obscured and Walther von Brauchitsch in front of the Armistice carriage The Compiègne Wagon was the train car...

Silk Road transmission of Buddhism

Part of the history of Buddhism in Asia Buddhist expansion in Asia: Mahayana Buddhism first entered the Chinese Empire (Han dynasty) through Silk Road during the Kushan Era. The overland and maritime Silk Roads were interlinked and complementary, forming what scholars have called the great circle of Buddhism.[1] Part of a series onChinese BuddhismChinese: Buddha History Buddhism in Central Asia Dharmaguptaka Silk Road transmission Dunhuang manuscripts Four Buddhist Persecutions in Chi...

2022 Italian general election

2022 Italian general election ← 2018 25 September 2022 Next → outgoing memberselected membersChamber of DeputiesAll 400 seats in the Chamber of Deputies201 seats needed for a majorityTurnout63.85% ( 9.09 pp) Party Leader % Seats +/– Centre-right coalition (237 seats) Brothers of Italy Giorgia Meloni 25.98 119 +87 Lega Matteo Salvini 8.79 66 −59 Forza Italia Silvio Berlusconi 8.11 45 −59 Us Moderates Maurizio Lupi 0.90 7 +4 Centre-left coalition (84 seats) Democrat...

هاماسكايين

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (مارس 2016) هاماسكايينArmenian All-National Cultural and Educational Society (بالإنجليزية)Համազգայինի Հայ Կրթական Եւ Մշակութային Միութիւն (با�...

Swedish Chef

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Swedish Chef – news · newspapers · books · scholar · JSTOR (November 2023) (Learn how and when to remove this message) Muppet character Børk redirects here. For other uses, see Bork (disambiguation). Fictional character The Swedish ChefThe Muppets characterFir...

Nick Knight (cricketer)

English cricketer This biography of a living person needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately from the article and its talk page, especially if potentially libelous.Find sources: Nick Knight cricketer – news · newspapers · books · scholar · JSTOR (June 2008) (Learn how and when to remove this message)...

Dennis-Yarmouth Regional High School

Public school in South Yarmouth, Massachusetts, United StatesDennis-Yarmouth Regional High SchoolAddress210 Station AvenueSouth Yarmouth, Massachusetts 02664United StatesInformationTypePublicOpen enrollment[1]Established1957PrincipalKendra BennettStaff79.53 (FTE)[2]Grades8–12Enrollment881 (2020–2021) [2]Student to teacher ratio11.08[2]CampusSuburbanColor(s)Green & White AthleticsMIAA - Division 2Athletics conferenceAtlantic Coast LeagueTeam...

Bad Arolsen

Bad Arolsen Schlossstrasse di pusat distrik Bad Arolsen - di sebelah barat Kirchplatz dengan gereja Lambang kebesaranLetak Bad Arolsen di Waldeck-Frankenberg NegaraJermanNegara bagianHessenWilayahKasselKreisWaldeck-FrankenbergPemerintahan • MayorJürgen van der Horst (Independen)Luas • Total126,32 km2 (4,877 sq mi)Ketinggian286 m (938 ft)Populasi (2013-12-31)[1] • Total15.368 • Kepadatan1,2/km2 (3,2/sq ...

PROFILPELAJAR.COM