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.

2. 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 ).

3. 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.