University of Wisconsin-Madison |
Math 541 |
Fall 2007
|
Let be a commutative ring. Define prime ideal and maximal ideal and give some examples.
Show that every field is an integral domain and give an example of an integral domain that is not a field.
Let be a commutative ring. Show that every maximal ideal is prime.
Give an example of a prime ideal that is not maximal.
Let be a ring (not necessarily commutative). Define maximal ideal.
Define simple ring.
Let be a ring (not necessarily commutative) and let be an ideal of . Show that is a maximal ideal of if and only if is a simple ring.
Show that every division ring is a simple ring.
Give an example of a simple ring that is not a division ring.
Show that if is a commutative simple ring then is a field.
Let be a commutative ring. Show that the two different definitions of maximal ideal are equivalent.