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A generalization of the notion of a [[Prime number|prime number]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744901.png" /> be an [[Integral domain|integral domain]] or commutative [[Semi-group|semi-group]] with an identity. A non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744902.png" /> that is not a divisor of unity is called prime if a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744903.png" /> can be divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744904.png" /> only if one of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744905.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744906.png" /> can be divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744907.png" />. Every prime element is irreducible, i.e. is divisible only by divisors of unity or elements associated to it. An irreducible element need not be prime; however, in a [[Gauss semi-group|Gauss semi-group]] both concepts coincide. Moreover, if every irreducible element of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744908.png" /> is prime, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p0744909.png" /> is a Gauss semi-group. Analogous statements hold for a [[Factorial ring|factorial ring]]. An element of a ring is prime if and only if the [[Principal ideal|principal ideal]] generated by this element is a [[Prime ideal|prime ideal]].
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A generalization of the notion of a [[Prime number|prime number]]. Let $G$ be an [[Integral domain|integral domain]] or commutative [[Semi-group|semi-group]] with an identity. A non-zero element $p\in G$ that is not a divisor of unity is called prime if a product $ab$ can be divided by $p$ only if one of the elements $a$ or $b$ can be divided by $p$. Every prime element is irreducible, i.e. is divisible only by divisors of unity or elements associated to it. An irreducible element need not be prime; however, in a [[Gauss semi-group|Gauss semi-group]] both concepts coincide. Moreover, if every irreducible element of a semi-group $G$ is prime, then $G$ is a Gauss semi-group. Analogous statements hold for a [[Factorial ring|factorial ring]]. An element of a ring is prime if and only if the [[Principal ideal|principal ideal]] generated by this element is a [[Prime ideal|prime ideal]].
  
 
There are generalizations of these notions to the non-commutative case (cf. [[#References|[2]]]).
 
There are generalizations of these notions to the non-commutative case (cf. [[#References|[2]]]).
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====Comments====
 
====Comments====
Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p07449010.png" /> in a commutative semi-group or in an integral domain are associates of each other if each is a divisor of the other; i.e., if there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p07449011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p07449012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074490/p07449013.png" />.
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Two elements $a,b$ in a commutative semi-group or in an integral domain are associates of each other if each is a divisor of the other; i.e., if there are $c,d$ such that $a=bc$, $b=ad$.

Revision as of 14:03, 19 March 2014

A generalization of the notion of a prime number. Let $G$ be an integral domain or commutative semi-group with an identity. A non-zero element $p\in G$ that is not a divisor of unity is called prime if a product $ab$ can be divided by $p$ only if one of the elements $a$ or $b$ can be divided by $p$. Every prime element is irreducible, i.e. is divisible only by divisors of unity or elements associated to it. An irreducible element need not be prime; however, in a Gauss semi-group both concepts coincide. Moreover, if every irreducible element of a semi-group $G$ is prime, then $G$ is a Gauss semi-group. Analogous statements hold for a factorial ring. An element of a ring is prime if and only if the principal ideal generated by this element is a prime ideal.

There are generalizations of these notions to the non-commutative case (cf. [2]).

References

[1] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[3] S. Lang, "Algebra" , Addison-Wesley (1974)


Comments

Two elements $a,b$ in a commutative semi-group or in an integral domain are associates of each other if each is a divisor of the other; i.e., if there are $c,d$ such that $a=bc$, $b=ad$.

How to Cite This Entry:
Prime element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_element&oldid=31393
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article