In mathematics, specifically ring theory, a principal ideal is an ideal ${\displaystyle I}$ in a ring ${\displaystyle R}$ that is generated by a single element ${\displaystyle a}$ of ${\displaystyle R}$ through multiplication by every element of ${\displaystyle R.}$ The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset ${\displaystyle P}$ generated by a single element ${\displaystyle x\in P,}$ which is to say the set of all elements less than or equal to ${\displaystyle x}$ in ${\displaystyle P.}$

The remainder of this article addresses the ring-theoretic concept.

While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under addition.

If ${\displaystyle R}$ is a commutative ring, then the above three notions are all the same. In that case, it is common to write the ideal generated by ${\displaystyle a}$ as ${\displaystyle \langle a\rangle }$ or ${\displaystyle (a).}$

A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.

As an example, ${\displaystyle \mathbb {Z} }$ is a principal ideal domain, which can be shown as follows. Suppose ${\displaystyle I=\langle n_{1},n_{2},\ldots \rangle }$ where ${\displaystyle n_{1}\neq 0,}$ and consider the surjective homomorphisms ${\displaystyle \mathbb {Z} /\langle n_{1}\rangle \rightarrow \mathbb {Z} /\langle n_{1},n_{2}\rangle \rightarrow \mathbb {Z} /\langle n_{1},n_{2},n_{3}\rangle \rightarrow \cdots .}$ Since ${\displaystyle \mathbb {Z} /\langle n_{1}\rangle }$ is finite, for sufficiently large ${\displaystyle k}$ we have ${\displaystyle \mathbb {Z} /\langle n_{1},n_{2},\ldots ,n_{k}\rangle =\mathbb {Z} /\langle n_{1},n_{2},\ldots ,n_{k+1}\rangle =\cdots .}$ Thus ${\displaystyle I=\langle n_{1},n_{2},\ldots ,n_{k}\rangle ,}$ which implies ${\displaystyle I}$ is always finitely generated. Since the ideal ${\displaystyle \langle a,b\rangle }$ generated by any integers ${\displaystyle a}$ and ${\displaystyle b}$ is exactly ${\displaystyle \langle \mathop {\mathrm {gcd} } (a,b)\rangle ,}$ by induction on the number of generators it follows that ${\displaystyle I}$ is principal.

Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define ${\displaystyle \gcd(a,b)}$ to be any generator of the ideal ${\displaystyle \langle a,b\rangle .}$

For a Dedekind domain ${\displaystyle R,}$ we may also ask, given a non-principal ideal ${\displaystyle I}$ of ${\displaystyle R,}$ whether there is some extension ${\displaystyle S}$ of ${\displaystyle R}$ such that the ideal of ${\displaystyle S}$ generated by ${\displaystyle I}$ is principal (said more loosely, ${\displaystyle I}$ becomes principal in ${\displaystyle S}$). This question arose in connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in number theory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others.