Homework (Fall 2014) # 5 Due Monday, Nov 24
                                   
                 


Hand-in (from Kaplansky)

  Sec. 2.3:  2, 3, 8, 11

Hint on 8: This problem is short but tricky.  You will need the following result which I have not proven in class: Let S' me a multiplicatively closed set in a ring R and let S be a multiplicatively closed subset of R that is contained in S'.  Then the image of S' in R_S is a multplicatively closed set in R_S. Moreover R_S' = (R_S)_S'.  You may use all this without proof (but try to see why it is true, it is not difficult).


Hint on 11: You need to first explain why R is Noetherian. This  
follows from a result we did a while back that says that if all the prime 
ideals of a ring are finitely generated, then R is Noetherian.  Then recall the proof of Theorem 97.  Also show that for ideals A and B, (AB)^{-1} contains A^{-1}B^{-1}.