Prime element

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A generalization of the notion of a prime number. Let be an integral domain or commutative semi-group with an identity. A non-zero element that is not a divisor of unity is called prime if a product can be divided by only if one of the elements or can be divided by . 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 is prime, then 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]).


[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)


Two elements 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 such that , .

How to Cite This Entry:
Prime element. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article