Prime element

In abstract algebra, an element $p$ of a commutative ring $R$ is said to be prime if it is not zero, not a unit and whenever $p$ divides $ab$ for some $a$ and $b$ in $R$ , then $p$ divides $a$ or $p$ divides $b$ . Equivalently, an element $p$ is prime if, and only if, the principal ideal $(p)$ generated by $p$ is a nonzero prime ideal.^[1]

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 numbers. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers.

Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible^[2] but the converse is not true in general. However, in unique factorization domains,^[3] or more generally in GCD domains, primes and irreducibles are the same.

Being prime is also 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[ $i$ ], the ring of Gaussian integers, since $2=i(1-i)^2$ and 2 does not divide any factor on the right.

Examples

The following are examples of prime elements in rings:

The integers ±2, ±3, ±5, ±7, ±11,... in the ring of integers Z
the complex numbers ( $1%2Bi$ ), 19, and ( $2%2B3i$ ) in the ring of Gaussian integers Z[ $i$ ]
the polynomials $x^2 - 2$ and $x^2%2B1$ in the ring of polynomials over Z.

References

Notes

^ Hungerford 1980, Theorem III.3.4(i), as indicated in the remark below the theorem and the proof, the result holds in full generality.
^ Hungerford 1980, Theorem III.3.4(iii)
^ Hungerford 1980, Remark after Definition III.3.5

Sources

Section III.3 of Hungerford, Thomas W. (1980), Algebra, Graduate Texts in Mathematics, 73 (Reprint of 1974 ed.), New York: Springer-Verlag, ISBN 978-0-387-90518-1, MR 0600654