TITLE: Is there a non-unital ring such that one of its quotient ring is unital?
QUESTION [7 upvotes]: Suppose $R$ is a non unital ring, but it has an ideal $I$ such that $R/I$ is unital, does this kind of animal exist?
This problem arouse when I try to show “$R$ is unital, $R/I$ is division ring, then $I$ is maximal. “ I am wondering whether the condition $R$ is unital can be dropped or not.
Thank you in advance.
By the way, is this kind of problem meaningful? I am pretty addicted to this kind of problems but it seems that nobody cares about it.

REPLY [3 votes]: Yes, it can happen. Consider e.g. $R:=\{p\in\Bbb Z[x]:p(0)\in 2\Bbb Z\}  $ and $I:=(2x-2,x^2-x)$. 
Then $2ax+I=2a+I$ for $a\in\Bbb Z$ and by induction, $x^n+I=x+I$, so $x+I$ will act as a multiplicative identity on $R/I$, and by the way, $R/I\cong\Bbb Z$.