In abstract algebra, a **principal ideal domain**, or **PID**, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If *x* and *y* are elements of a PID without common divisors, then every element of the PID can be written in the form *ax* + *by*.

Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind rings. All Euclidean domains and all fields are principal ideal domains.

**Commutative rings**⊃**integral domains**⊃**integrally closed domains**⊃**unique factorization domains**⊃**principal ideal domains**⊃**Euclidean domains**⊃**fields**

Read more about Principal Ideal Domain: Examples, Modules, Properties

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