Prime Number - Generalizations - Prime Elements in Rings

Prime Elements in Rings

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is

In any ring R, any prime element is irreducible. The converse does not hold in general, but does hold for unique factorization domains.

The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z, that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z, 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.