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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 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 rings and all fields
are principal ideal domains.
- K: any field,
- Z: the ring of integers,
- K[x]: rings of polynomials in one variable with
coefficients in a field. (The converse is also true; that is, if
A[x] is a PID, then A is a field.)
Furthermore, a ring of formal power series over a field is a PID
since every ideal is of the form (xk).
- Z[i]: the ring of Gaussian integers
- Z[ω] (where ω is a cube root of 1): the Eisenstein integers
Examples of integral domains that are not PIDs:
- Z[x]: the ring 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.
- K[x,y]: The ideal
(x,y) is not principal.
The key result is the structure theorem: If R is a
principal ideal domain, and M is a finitely generated
R-module, then M is
a direct sum of cyclic modules, i.e., modules with one generator.
The cyclic modules are isomorphic to R / xR for some
If M is a free module over a principal ideal domain
R, then every submodule of M is again free. This
does not hold for modules over arbitrary rings, as the example
of modules over
In a principal ideal domain, any two elements
a,b have a greatest common divisor, which
may be obtained as a generator of the ideal (a,b).
All Euclidean domains are principal ideal
domains, but the converse is not true. An example of a principal
ideal domain that is not a Euclidean domain is the ring
Every principal ideal domain is a unique factorization domain
converse does not hold since for any field K,
K[X,Y] is a UFD but is not a PID (to
prove this look at the ideal generated by
It is not the whole ring since it contains no polynomials of degree
0, but it cannot be generated by any one single element).
- Every principal ideal domain is Noetherian.
- In all unital rings, maximal ideals are prime. In principal ideal domains a near
converse holds: every nonzero prime ideal is maximal.
- 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
Let A be an integral domain. Then the following are
- A is a PID.
- Every prime ideal of A is principal.
- A is a Dedekind domain that is a UFD.
- Every finitely generated ideal of A is principal
(i.e., A is a Bézout domain) and A satisfies
chain condition on principal ideals.
- A admits a Dedekind–Hasse norm.
A field norm is a
Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a
PID. (4) compares to:
- An integral domain is a UFD if and only if it is a GCD domain (i.e., a
domain where every two elements has a greatest common divisor)
satisfying the ascending chain condition on principal ideals.
An integral domain is a Bézout domain if and only if any two
elements in it has a gcd that is a linear combination of the
two. A Bézout domain is thus a GCD domain, and (4) gives yet
another proof that a PID is a UFD.
See Fraleigh & Katz (1967), p. 73, Corollary of Theorem 1.7,
and notes at p. 369, after the corolary of Theorem 7.2
See Fraleigh & Katz (1967), p. 385, Theorem 7.8 and p. 377,
See also Ribenboim (2001), p. 113, proof of lemma
Wilson, Jack C. "A Principal Ring that is Not a Euclidean Ring."
Math. Mag 46 (Jan 1973) 34-38 
George Bergman, A principal ideal domain that is not Euclidean
- developed as a series of exercises PostScript file
Proof: every prime ideal is generated by one element, which is
necessarily prime. Now refer to the fact that an integral domain is
a UFD if and only if its prime ideals contain prime elements.
Jacobson (2009), p. 148, Theorem 2.23.
Fraleigh & Katz (1967), p. 368, Theorem 7.2
Hazewinkel, Gubareni & Kirichenko (2004), p.166, Theorem 7.2.1.
T. Y. Lam and Manuel L. Reyes,
A Prime Ideal Principle in Commutative Algebra
- Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko.
Algebras, rings and modules. Kluwer
Academic Publishers, 2004. ISBN 1-4020-2690-0
- John B. Fraleigh, Victor J. Katz. A first course in
abstract algebra. Addison-Wesley Publishing Company. 5 ed.,
1967. ISBN 0-201-53467-3
Jacobson. Basic Algebra I. Dover, 2009. ISBN
- Paulo Ribenboim. Classical theory of algebraic
numbers. Springer, 2001. ISBN 0387950702