TITLE: Does no zero divisors in quotient ring imply no zero divisors in original ring?
QUESTION [5 upvotes]: Let $R$ be a ring and let $I$ be a proper ideal of $R$.
If $R/I$ has no zero divisors, then is it true that $R$ has no zero divisors?
My attempt:
Suppose $R$ has zero divisors, say $ab=0$ for some $a,b\in R^*$. Then $(a+I)(b+I)=ab+I=I$. However, I cannot exclude the case where $a,b\in I$.
Any ideas?

REPLY [2 votes]: No.
Z6/<2> is an integral domain as Z6 is a CRU and <2> is a prime ideal but Z6 is not a integral domain.