TITLE: which of the following statements are true in ring theory
QUESTION [5 upvotes]: which of the following statements are true in ring theory?   

(a) Let $R$ is a commutative ring with unity and $I$ be an ideal. Then $R/I$ is an integral domain.
  (b) If $R$ is a commutative ring with unity the units of $R[x]$ are the units in $R$.

(a) I think it is not true as $R/I$ is an integral domain iff I is an prime ideal but I have no counter example.
(b) I have no idea.  
can I get some help.Thanks for your cooperation.

REPLY [2 votes]: (a) Which ideals are prime in $\mathbb Z$? 
(b) Hint: $(1-aX)(1+aX)=1-a^2X^2$.
What happens if $R$ is a ring and $a \in R$ is such that $a^2=0$? Can you think of any such ring?