Chapter 5.2, Problem 19E

a. Give an example where $a$ and $b$ are not zero divisors in a ring $R$, but the sum $a+b$ is a zero divisor.

Give an example where $a$ and $b$ are zero divisors in a ring $R$ with $a+b\ne 0$, and $a+b$ is not a zero divisor.

Prove that the set of all elements in a ring $R$ that are not zero divisors is closed under multiplication.