Math 724

Fall 2001

Exam 1

 

You may use any text and me. Due Friday October 19, 2001.

 

 

 

  1. Let  be a Noetherian ring. Show that the power series ring  is also Noetherian.

 

  1. Let  be an integral domain and  a multiplicatively closed subset of  ().

a)      Show that

b)      (Lamentations) In this part we show that part a) almost never works for power series. Show that if  is a UFD and is a multiplicatively closed subset of  then  but equality holds if and only if  (and so the localization is superfluous).

c)      Give an example of a ring  and an ideal  such that  is strictly contained in the ideal  (This is contrary to the polynomial situation where ).

 

  1. Let  be a ring in which every ideal may be generated by a single element (such a ring is called a principal ideal ring or PIR). Show that any PIR has dimension at most 1 (that is, there is no chain of three prime ideals). Give an example of a zero-dimensional PIR (not a field) and a zero-dimensional ring that is not a PIR.