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

Last update: 01 February 2012


Let R be a ring.

  1. Let R be an integral domain. If R is a UFD or R is quasilocal then Pic(R)=0.
  2. Let R be an algebraic number ring. If R is a PID then Cl(R)=0.


(Nagata's theorem.) This relates the class group Cl(R) of R and the class group Cl(RS) of a localization of R.

Example. Let D be an integral domain and let 𝕂 be the field of fractions of D. Let R=D x2 x3 . Then Pic(𝕂 x2 x3 )=𝕂. See Lam's book.

Notes and References

Where are these from?



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