§2T. Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains

§2T. Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 08 March 2011

Euclidean domains, principal ideal domains, and unique factorization domains

R is a Euclidean domain R is a PID.


A Euclidean domain is a principal ideal domain.

R is a PID R is a UFD.

Definition. Let R be an integral domain.

The following proposition shows that every property of divisors can be written in terms of containments of ideals and vice versa.

Let p,qR and p and q denote the ideals generated by the elements p and q respectively. Then

  1. p is a unit p=R.
  2. p divides qqp.
  3. p is a proper divisor of qqp R.
  4. p is an associate of qq=p.
  5. p is irreducible
    1. p0,
    2. pR,
    3. If qR and qp then either q=p or q=R.


A principal ideal domain is a unique factorization domain.

The proof of Theorem 1.3 will require the following lemmas.

If R is a principal ideal domain and pR is an irreducible element of R then p is a prime ideal.

Let R be a principal ideal domain. There does not exist an infinite sequence of elements a1,a2,R such that 0 a1 a2

Greatest common divisors


Let R be a unique factorization domain and let a0,a1,,an R. Then

  1. gcda0,a1,,an exists.
  2. gcda0,a1,,an is unique up to multiplication by a unit.


[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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