Question 1: Elements of

  1. What are the units in ?
  2. What are the nilpotent elements of ?
  3. What are the zero divisors in ?

(3 points)

Question 2: Zero divisors in

Let be a nontrivial ring, and let . Prove that if is not a zero divisor in , then is not a zero divisor in . (Hint: If you’re stuck, warm-up with a simple case like )

Note (nothing to prove here):

As a consequence of this question, we see that if is an integral domain, then so is .

(3 Points)

Question 3: Counting homomorphisms

How many different homomorphisms are there when

  1. and
  2. and
  3. and
  4. and

(4 points)