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

 

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

 

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

a)  is a valuation domain.

b)  is a quasi-local Bezout domain.