Prime element

In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general.

An element p of a commutative ring R is said to be prime if it is not zero or 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. (Note that in an integral domain, the ideal (0) is a prime ideal, but 0 is an exception in the definition of 'prime element'.)

Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each nonzero 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.

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