where u is a unit and p1, p2,. . . ,pk are primes in R. Note that the factorization is essentially unique (by the same argument used to prove uniqueness of factorization in PIDs). Note also that if R is a UFD, any finite collection a1, . . . , an ∈ R has a highest common factor. For we can take out prime factors until we write ai = rbi where the b1, . . . , bn have no proper factors in common. Then r is the (unique up to units) highest common factor. We write r = hcf(a1, . . . , an), but note that unless R is a PID we will not in general have r = λ1a1 + . . . + λnan for λi ∈ R. Observe that if R is an integral domain then R is a UFD iff it satisfies the following condition:
[1]
M. Nagata.
A Jacobian criterion of simple points
,
1957
.
[2]
C. Chevalley,et al.
Introduction to the theory of algebraic functions of one variable
,
1951
.
[3]
D. Buchsbaum,et al.
UNIQUE FACTORIZATION IN REGULAR LOCAL RINGS.
,
1959,
Proceedings of the National Academy of Sciences of the United States of America.
[4]
M. Nagata.
A remark on the unique factorization theorem
,
1957
.
[5]
F. Severi.
Sulla varietà che rappresenta gli spazî subordinati di data dimensione, immersi in uno spazio lineare
,
1915
.
[6]
M. Noether.
Zur Grundlegung der Theorie der algebraischen Raumcurven.
,
1882
.