Math 720

Fall 2000

Homework 6

1. Let R be a PID.

a) Show that every homomorphic image of R is a PIR.

b) Show that R is 1-dimensional (which means that every non-zero prime ideal is maximal).

c) Show that if a homomorphic image of R (say D) is an integral domain then either D is isomorphic to R or D is a field.

2. Let R be an integral domain and let S be a multiplicatively closed subset of R (which means that ) Assume that 0 is not an element of S.

a) Show that there is an ideal (I) maximal with respect to the exclusion of S (that is,

b) Show that I is prime.

3. Let R be commutative with 1. We recall that the radical of I, rad(I)= is the set of all such that for some n.

a) Show that rad(I) is an ideal of R.

b) Show that R/rad(I) has no nonzero nilpotents.

c) Show that rad(I)=

d) Show that the set of nilpotent elements of R form an ideal and this ideal is the intersection of all prime ideals in R.

4. Let R be commutative with 1. Show that the following are equivalent:

a) R has a unique prime ideal.

b) Every nonunit of R is nilpotent.

c) R has a minimal prime ideal which contains all zero divisors, and all nonunits of R are zero-divisors.