principal ideal domain

In some rings ideals can be complicated, generated by many elements at once. In a principal ideal domain, by contrast, every ideal is as simple as possible: it is the set of all multiples of a single element. The whole ideal theory collapses to the study of divisibility among elements, which is why PIDs feel so much like the integers.

Formally, a principal ideal domain is an integral domain R in which every ideal I has the form (a) = a·R = { a·r : r in R } for some a in R. Equivalently, every ideal is generated by one element. Such rings are automatically Noetherian and are unique factorization domains, and in fact they are precisely the integral domains that are both UFDs and have Krull dimension at most one (every nonzero prime ideal is maximal).

The reason PIDs matter beyond their elegance is the structure theorem for finitely generated modules over a PID: every such module is a direct sum of cyclic modules R/(d_i) and free parts. This single theorem yields both the classification of finitely generated abelian groups (over Z) and the rational and Jordan canonical forms of a linear operator (over k[x]). Be careful: a polynomial ring in two variables k[x, y] is not a PID — the ideal (x, y) genuinely needs two generators.