Integral Domain - Divisibility, Prime Elements, and Irreducible Elements

Divisibility, Prime Elements, and Irreducible Elements

If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.

The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.

If a divides b and b divides a, then we say a and b are associated elements or associates.

If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.

If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalent, an element is prime if and only if an ideal generated by it is a nonzero prime ideal. Every prime element is irreducible. Conversely, in a GCD domain (e.g., a unique factorization domain), an irreducible element is a prime element.

The notion of prime element generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. While every prime is irreducible, the converse is not in general true. For example, in the quadratic integer ring the number 3 is irreducible, but is not a prime because 9, the norm of 3, can be factored in two ways in the ring, namely, and . Thus, but 3 does not divide nor The numbers 3 and are irreducible as there is no where or as has no integer solution.

While unique factorization does not hold in the above example, if we use ideals we do get unique factorization. See Lasker–Noether theorem.