Question 1: Elements of
- What are the units in ?
- What are the nilpotent elements of ?
- What are the zero divisors in ?
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 .
Question 3: Counting homomorphisms
How many different homomorphisms are there when