Last update: 25 June 2012
Let be a commutative ring. Let be a finitely generated projective module. Let
- The rank of at , is the rank of the free module
- The rank of is if
is an abelian group.
Let be a commutative ring, a multiplicatively closed subset such that if then is not a zero divisor in
An invertible sub module is a sub module of such that there exists a sub module of such that
with product multiplication. Then there is an exact sequence of abelian groups
If is an integral domain,
and is the field of fractions of then
- An invertible fractional ideal is an element of
- A principal fractional ideal is an element of
Notes and References
The Theorem is given in [Bou, Comm. Ch.II, §5 no.4, Proposition 7] and the definitions and proofs of well-definedness of rank of are given in [Bou, Comm. Ch.II, §5 no.3].