If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain:
proof
$a$ not  prime, therefore:
$a \mid bc$ and $a\nmid b$ and $a\nmid c$
Therefore $b+aA \neq aA$ and $c+aA \neq aA$
But (b+aA)(c+aA) = aA
According to my professor that concludes the proof
My question is: if $b+aA = aA$ or $c+aA = aA$ would they be the absorbing element of $A/aA$?
In that case, the proof would make perfect sense to me

The absorbing (or zero) element of $A/aA$ is $aA$.
Therefore if $b+aA=aA$ you just get the trivial equality $0.(c+aA)=0$ in $A/aA$.
