Math 720

Fall 2000

Homework 5

1. Prove that if R is a finite commutative ring then any element of R is either a zero-divisor or a unit. Use this to prove the general statement “if R is a finite commutative ring with no nonzero zero-divisors, then R is a field.” (Note: I did not say that R has an identity…you must show this).

2. We say that the element is nilpotent if for some integer .

a) Show that if then is nilpotent.

b) Give an example of two nilpotents whose sum is not nilpotent.

c) Show that if R has an identity and a is nilpotent, then 1+a is a unit.

d) Nilpotents are zero divisors. Is the converse true?

3. Let R be a ring such that for all Show that R is necessarily commutative. Does R necessarily have an identity? Give a finite and an infinite example of such a ring.

4. We say that the ring R has characteristic n if nr = 0 for all r in R (and n is the smallest such positive integer). If no such integer exists, then we say the characteristic of R is 0.

a) Show that if R is an integral domain, then the characteristic of R is either 0 or prime.

b) Show that if R is a ring such that for all then the characteristic of R divides (hence is nonzero).

5. Let R be a commutative ring with 1 and let the characteristic of R be the prime number Show that for all Use this to show that the map given by is a ring homomorphism.