Principal ideal domain

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For a similar structure that is not necessarily commutative and may have zero-divisors, see Principal ideal ring.

In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some authors such as Bourbaki refer to PIDs as principal rings.

Principal ideal domains are mathematical objects that behave like the integers, with respect to divisibility: any element of a PID has a unique factorization 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, etc.

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

Some authors additionally require principal ideal domains to have Krull dimension one, or equivalently, to possess ideals that are nonzero and proper, in order to specifically exclude fields,[1] while most other sources regard fields as trivial special cases of principal ideal domains.

Principal ideal domains appear in the following chain of class inclusions: Template:Commutative ring classes Template:Algebraic structures

Examples

Examples include:

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} : any field, whose only ideals are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} ,
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} : the ring of integers,[2]
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K[x]} : rings of polynomials in one variable with coefficients in a field. (The converse is also true, i.e. if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A[x]} is a PID then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x^k)} ,
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[i]} : the ring of Gaussian integers,[3]
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[\omega]} (where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} is a primitive cube root of 1): the Eisenstein integers,
  • Any discrete valuation ring, for instance the ring of p-adic integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_p} .

Non-examples

Examples of integral domains that are not PIDs:

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[\sqrt{-3}]} is an example of a ring that is not a unique factorization domain, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 = 2\cdot 2 = (1+\sqrt{-3})(1-\sqrt{-3}).} Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains. Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 2, 1+\sqrt{-3} \rangle} is an ideal that cannot be generated by a single element.
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[x]} : the ring of all polynomials with integer coefficients. It is not principal because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 2, x \rangle} is an ideal that cannot be generated by a single polynomial.
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K[x_1, x_2, \ldots, x_n],} the ring of polynomials in at least two variables over a ring K is not principal, since the ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x_1, x_2 \rangle} is not principal.
  • Most rings of algebraic integers are not principal ideal domains. This is one of the main motivations behind Dedekind's definition of Dedekind domains, which allows replacing unique factorization of elements with unique factorization of ideals. In particular, many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[\zeta_p],} for the primitive p-th root of unity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta_p,} are not principal ideal domains.[4] The class number of a ring of algebraic integers gives a measure of "how far away" the ring is from being a principal ideal domain.

Modules

Main article: Structure theorem for finitely generated modules over a principal ideal domain

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/xR} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in R} [5] (notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} may be equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} , in which case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/xR} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ).

If M is a free module over a principal ideal domain R, then every submodule of M is again free.[6] This does not hold for modules over arbitrary rings, as the example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2,X) \subseteq \mathbb{Z}[X]} of modules over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[X]} shows.

Properties

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}\bigl[\tfrac12\bigl(1+\sqrt{-19}~\!\bigr)\bigr]} ,[7][8] this was proved by Theodore Motzkin and was the first case known.[9] In this domain no q and r exist, with 0 ≤ |r| < 4, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl(1+\sqrt{-19~\!}\bigr)=(4)q+r} , despite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+\sqrt{-19}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} having a greatest common divisor of 2.

Every principal ideal domain is a unique factorization domain (UFD).[10][11][12][13] The converse does not hold since for any UFD K, the ring K[X, Y] of polynomials in 2 variables is a UFD but is not a PID. (To prove this look at the ideal generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle X,Y \right\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 unital 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.

Let A be an integral domain, the following are equivalent.

  1. A is a PID.
  2. Every prime ideal of A is principal.[14]
  3. A is a Dedekind domain that is a UFD.
  4. Every finitely generated ideal of A is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain condition on principal ideals.
  5. A admits a Dedekind–Hasse norm.[15]

Any Euclidean 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 have 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 have 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 also

Notes

  1. ↑ Lorenzini, Dino (1996). An invitation to arithmetic geometry. Graduate studies in mathematics. Providence, Rhode Island: American Mathematical Society. ISBN 978-1-4704-2069-7.
  2. ↑ See Fraleigh & Katz (1967), p. 73, Corollary of Theorem 1.7, and notes at p. 369, after the corollary of Theorem 7.2
  3. ↑ See Fraleigh & Katz (1967), p. 385, Theorem 7.8 and p. 377, Theorem 7.4.
  4. ↑ Milne, James. "Algebraic Number Theory" (PDF). p. 5.
  5. ↑ See also Ribenboim (2001), p. 113, proof of lemma 2.
  6. ↑ Lecture 1. Submodules of Free Modules over a PID math.sc.edu Retrieved 31 March 2023
  7. ↑ Wilson, Jack C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag 46 (Jan 1973) 34-38 [1]
  8. ↑ George Bergman, A principal ideal domain that is not Euclidean - developed as a series of exercises PostScript file
  9. ↑ Motzkin, Th (December 1949). "The Euclidean algorithm". Bulletin of the American Mathematical Society. 55 (12): 1142–1146. doi:10.1090/S0002-9904-1949-09344-8. ISSN 0002-9904.
  10. ↑ 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.
  11. ↑ Jacobson (2009), p. 148, Theorem 2.23.
  12. ↑ Fraleigh & Katz (1967), p. 368, Theorem 7.2
  13. ↑ Hazewinkel, Gubareni & Kirichenko (2004), p.166, Theorem 7.2.1.
  14. ↑ T. Y. Lam; Manuel L. Reyes. "A Prime Ideal Principle in Commutative Algebra" (PDF). Archived from the original (PDF) on 26 July 2010. Retrieved 31 March 2023.
  15. ↑ Hazewinkel, Gubareni & Kirichenko (2004), p.170, Proposition 7.3.3.

References

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