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 $I$ there is an element $i\in I$ such that $⟨i⟩=I$; equivalently, every element of $I$ is a multiple of $i$.

Since ideals are kernels of ring homomorphisms (proof), this is saying that a PID $R$ has the special property that every ring homomorphism from $R$ 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 $Z$ 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 $\left\{0\right\}$ (i.e. generated by $0$) or else is the entire ring (i.e. generated by $1$).
• The ring $F\left[X\right]$ of polynomials over a field $F$ is a PID, because it is a Euclidean domain. (Its Euclidean function is "take the degree of the polynomial".)
• The ring of Gaussian integers, $Z\left[i\right]$, is a PID because it is a Euclidean domain. (Proof; its Euclidean function is "take the norm".)
• The ring $Z\left[X\right]$ (of integer-coefficient polynomials) is not a PID, because the ideal $⟨2,X⟩$ is not principal. This is an example of a unique_factorisation_domain which is not a PID.
• The ring $Z_6$ is not a PID, because it is not an integral domain. (Indeed, $3\times 2=0$ in this ring.)

There are examples of PIDs which are not Euclidean domains, but they are mostly uninteresting. One such ring is $Z\left[\frac{1}{2}\right(1+\sqrt{-19}\left)\right]$. (Proof.)

Properties

• Every PID is a unique_factorisation_domain. (Proof; this fact is not trivial.) The converse is false; see the case $Z\left[X\right]$ 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.