# Prime element

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