# Principal ideal domain

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 $$\langle i \rangle = I$$; 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 $$\mathbb{Z}$$ 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 $$\{ 0 \}$$ (i.e. generated by $$0$$) or else is the entire ring (i.e. generated by $$1$$).

• The ring $$F[X]$$ 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, $$\mathbb{Z}[i]$$, is a PID because it is a Euclidean domain. (Proof; its Euclidean function is “take the norm”.)

• The ring $$\mathbb{Z}[X]$$ (of integer-coefficient polynomials) is not a PID, because the ideal $$\langle 2, X \rangle$$ is not principal. This is an example of a unique factorisation domain which is not a PID. proof of this

• The ring $$\mathbb{Z}_6$$ 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 $$\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]$$. (Proof.)

# Properties

• Every PID is a unique factorisation domain. (Proof; this fact is not trivial.) The converse is false; see the case $$\mathbb{Z}[X]$$ 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.

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