In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).

Examples are the ring of integers, all fields, and rings of polynomials in one variable with coefficients in a field. All Euclidean domains are principal ideal domains, but the converse is not true.

An example of an integral domain that is not a PID is the ring Z^* of all polynomials with integer coefficients. It is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.

Properties

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In a principal ideal domain, any two elements have a greatest common divisor, and almost always have more than one.

Every principal ideal domain is a unique factorization domain (UFD). The converse does not hold since for any field K, K^* is a UFD but is not a PID (to prove this look at the ideal generated by $\backslash left\backslash langle\; X,Y\; \backslash right\backslash rangle$. It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element).

1. Every principal ideal domain is Noetherian.
2. In all rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
3. All principal ideal domains are integrally closed.

The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.

So that PID $\backslash subseteq$ Dedekind $\backslash cap$ UFD . However there is another theorem which states that any unique factorisation domain that is a Dedekind domain is also a principal ideal domain. Thus we get the reverse inclusion Dedekind $\backslash cap$UFD $\backslash subseteq$PID, but then this shows equality and hence, Dedekind $\backslash cap$ UFD = PID.

An example of a principal ideal domain that is not a Euclidean domain is the ring $\backslash Bbb\{Z\}\backslash left*$ (Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973). Commutative algebra

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