# Question 1: Elements of

- What are the units in ?
- What are the nilpotent elements of ?
- 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 youâ€™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

- and
- and
- and
- and

(4 points)