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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. [[#References|[2]]]).

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



====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 $a=bc$, $b=ad$.