Good example of a singular quotient of modules that isn't a quotient of an essential extension, take 2 What is a good example of a quotient of right $R$ modules $B/A$ such that $B/A$ is singular, but $A$ is not essential in $B$, and $B$ is directly irreducible?

*

*singular means the annihilator of each element is an essential right ideal of $R$, and

*essential means that $A$ intersects all nonzero submodules of $B$ nontrivially, and

*directly irreducible means, of course, it cannot be expressed as a direct sum of two proper submodules.

My earlier version of this question turned out to have a simple solution using a clever trick. I'm not usually one to move goalposts, but in this case I was really hoping to be introduced to something a bit more interesting, hence the irreducibility hypothesis has been added to push harder.
Adding the criterion does not seem to be enough to create a proof that $A$ is essential in $B$, so I think there's still probably a counterexample.
 A: The formulation of the question allows $A=0$. Restricted to this subcase, the only thing needed is that $B$ is nontrivial, singular, and not directly decomposable. For example, let $R=\mathbb Z$ and $B=\mathbb Z_2$.
A: Let $R = \mathbb Z \langle x,y,z\rangle$ be the free ring of rank $3$.
Let $B$ be the $R$-module whose underlying additive group is  $V=\mathbb Z_2\oplus \mathbb Z_2, \oplus \mathbb Z_2$. Make $B$ an $R$-module by defining operators $x, y, z$ on $V$ as follows:
$x((a,b,c))=(a,0,0)$, 
$y((a,b,c)) = (0,0,c)$, 
$z((a,b,c)) = (b,0,b)$.
This module has five submodules, $0, A=R(1,0,0), C=R(0,0,1), A+C, B$.

Any element of a finite $R$-module is singular, since $R$ is free. All modules here are finite (including $B/A$), hence singular. $A$ is not an essential submodule of $B$, since $C\leq B$ is nontrivial and disjoint from $A$. $B$ is directly indecomposable since $B$ is a join irreducible element of its submodule lattice. This example differs from the other one I put on this page in that all conditions are satisfied, but $A\neq 0$.
