# Question 1: Elements of $\mathbb{Z}\times \mathbb{Z}$

1. What are the units in $\mathbb{Z}\times\mathbb{Z}$?
2. What are the nilpotent elements of $\mathbb{Z}\times\mathbb{Z}$?
3. What are the zero divisors in $\mathbb{Z}\times\mathbb{Z}$?

(3 points)

# Question 2: Zero divisors in $R[x]$

Let $R$ be a nontrivial ring, and let $f=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0\in R[x]$. Prove that if $a_n$ is not a zero divisor in $R$, then $f$ is not a zero divisor in $R[x]$. (Hint: If you’re stuck, warm-up with a simple case like $f=x^2-1$)

## Note (nothing to prove here):

As a consequence of this question, we see that if $R$ is an integral domain, then so is $R[x]$.

(3 Points)

# Question 3: Counting homomorphisms

How many different homomorphisms $\varphi:R\to S$ are there when

1. $R=\mathbb{Z}$ and $S=\mathbb{Z}[x]$
2. $R=\mathbb{Z}/7$ and $S=\mathbb{Z}/49$
3. $R=\mathbb{Z}/14$ and $S=\mathbb{Z}/7$
4. $R=\mathbb{Z}\times\mathbb{Z}$ and $S=\mathbb{Z}/12$

(4 points)