Math 724

Fall 2001

Homework 3

 

Due Monday, November 5, 2001.

 

  1. We say that the ring extension  is a survival extension if given any ideal ,  We say the extension  is a survival pair if given any intermediate extensions , then is a survival extension (similarly  is a lying over pair if given any  then  is LO).

a)      Show that any lying over pair is a survival pair (the converse is also true but much harder).

b)      Let be a domain. Show that  is a lying over pair if and only ifis a field (hint: consider the intermediate extension  where  is a nonzero nonunit).

 

  1. Show that the extension  is LO and GD, but not INC. Show that the extension fails to be GU if

 

  1. Let  be a ring extension and  a minimal prime ideal. Prove that  is lain over by a prime ideal of