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| In mathematics, specifically in [[abstract algebra]], a '''prime element''' of a [[commutative ring]] is an object satisfying certain properties similar to the [[prime number]]s in the integers and to [[irreducible polynomial]]s. Care should be taken to distinguish prime elements from [[irreducible element]]s, a similar concept which is the same in many rings.
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| ==Definition==
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| An element <math>p</math> of a [[commutative ring]] <math>R</math> is said to be '''prime''' if it is not zero or a [[unit (ring theory)|unit]] and whenever <math>p</math> [[Divisibility_(ring_theory)|divides]] <math>ab</math> for some <math>a</math> and <math>b</math> in <math>R</math>, then <math>p</math> divides <math>a</math> or <math>p</math> divides <math>b</math>. Equivalently, an element <math>p</math> is prime if, and only if, the [[principal ideal]] <math>(p)</math> generated by <math>p</math> is a nonzero [[prime ideal]].<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(i)}}, as indicated in the remark below the theorem and the proof, the result holds in full generality.</ref>
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| Interest in prime elements comes from the [[Fundamental theorem of arithmetic]], which asserts that each [[integer]] can be written in essentially only one way as 1 or −1 multiplied by a product of positive [[prime number]]s. This led to the study of [[unique factorization domain]]s, which generalize what was just illustrated in the integers.
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| Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in '''Z''' but it is not in '''Z'''[<math>i</math>], the ring of [[Gaussian integers]], since <math>2=(1+i)(1-i)</math> and 2 does not divide any factor on the right.
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| ==Connection with prime ideals==
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| {{Main|Prime ideal}}
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| An ideal ''I'' in the ring ''R'' (with unity) is [[prime ideal|prime]] if the factor ring ''R''/''I'' is an [[integral domain]].
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| A nonzero [[principal ideal]] is [[prime ideal|prime]] if and only if it is generated by a prime element.
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| ==Irreducible elements==
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| {{Main|Irreducible element}}
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| Prime elements should not be confused with [[irreducible element]]s. In an [[integral domain]], every prime is irreducible<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(iii)}}</ref> but the converse is not true in general. However, in unique factorization domains,<ref>{{harvnb|Hungerford|1980|loc=Remark after Definition III.3.5}}</ref> or more generally in [[GCD domain]]s, primes and irreducibles are the same.
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| ==Examples==
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| The following are examples of prime elements in rings:
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| * The integers ±2, ±3, ±5, ±7, ±11,... in the [[ring of integers]] '''Z'''
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| * the complex numbers (<math>1+i</math>), 19, and (<math>2+3i</math>) in the ring of [[Gaussian integers]] '''Z'''[<math>i</math>]
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| * the polynomials <math>x^2 - 2</math> and <math>x^2+1</math> in the [[ring of polynomials]] over '''Z'''.
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| ==References==
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| ;Notes
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| {{reflist}}
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| ;Sources
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| *Section III.3 of {{Citation
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| | authorlink=Thomas W. Hungerford
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| | last=Hungerford
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| | first=Thomas W.
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| | title=Algebra
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| | edition=Reprint of 1974
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| | year=1980
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| | publisher=[[Springer-Verlag]]
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| | location=New York
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| | series=Graduate Texts in Mathematics
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| | volume=73
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| | isbn=978-0-387-90518-1
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| | mr=0600654
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| }}
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| *{{citation |author=Jacobson, Nathan |title=Basic algebra. II |edition=2 |publisher=W. H. Freeman and Company |place=New York |year=1989 |pages=xviii+686 |isbn=0-7167-1933-9 |mr=1009787}}
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| *{{citation |author=Kaplansky, Irving |title=Commutative rings |publisher=Allyn and Bacon Inc. |place=Boston, Mass. |year=1970 |pages=x+180 |mr=0254021 }}
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| [[Category:Ring theory]]
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Hello! My name is Maryanne.
It is a little about myself: I live in France, my city of Torcy.
It's called often Eastern or cultural capital of . I've married 2 years ago.
I have two children - a son (Lynda) and the daughter (Aileen). We all like Baton twirling.
Feel free to visit my homepage :: on the spot trans 7