In ring theory, an integral domain is a principal ideal domain (or PID) if every ideal can be generated by a single element. That is, for every ideal there is an element such that ; equivalently, every element of is a multiple of .
Since ideals are kernels of ring homomorphisms (proof), this is saying that a PID has the special property that every ring homomorphism from acts “nearly non-trivially”, in that the collection of things it sends to the identity is just “one particular element, and everything that is forced by that, but nothing else”.
Examples
Every Euclidean domain is a PID. (Proof.)
Therefore is a PID, because it is a Euclidean domain. (Its Euclidean function is “take the modulus”.)
Every field is a PID because every ideal is either the singleton (i.e. generated by ) or else is the entire ring (i.e. generated by ).
The ring of polynomials over a field is a PID, because it is a Euclidean domain. (Its Euclidean function is “take the degree of the polynomial”.)
The ring of Gaussian integers, , is a PID because it is a Euclidean domain. (Proof; its Euclidean function is “take the norm”.)
The ring (of integer-coefficient polynomials) is not a PID, because the ideal is not principal. This is an example of a unique_factorisation_domain which is not a PID.
proof of thisThe ring is not a PID, because it is not an integral domain. (Indeed, in this ring.)
There are examples of PIDs which are not Euclidean domains, but they are mostly uninteresting. One such ring is . (Proof.)
Properties
Every PID is a unique_factorisation_domain. (Proof; this fact is not trivial.) The converse is false; see the case above.
In a PID, “prime” and “irreducible” coincide. (Proof.) This fact also characterises the maximal ideals of PIDs.
Every PID is trivially Noetherian: every ideal is not just finitely generated, but generated by a single element.