Math 724

Fall 2001

Exam 2

Due Friday November 16, 2001.

1. Let be an integral domain and a fractional ideal of

a) Show that is invertible if and only if is a projective module.

b) Show that if is invertible, it is rank one, that is, there is an module such that

2. Let be a Noetherian domain with quotient field Show that an element is integral over if and only if it is almost integral over

3. Let be a domain. Show that the following conditions are equivalent:

a) is a valuation domain.

b) is a quasi-local Bezout domain.