Intersection of cyclic submodules and ideals

Question

Let $A$ be a ring and $L$ be an $A$-module. Let $l_1, l_2\in L$ and consider the respective cyclic submodules $Al_i\cong A/\operatorname{ann}(l_i)$. Can the intersection $Al_1\cap Al_2$ be written as a quotient of $A$ be some suitable ideal?

Answer 1

The answer is no, as shown by the following counterexample.

Consider the three dimensional algebra $A=K[x,y]/(xy)$, the four dimensional $A$-module $L$ given by the representation $A\to\mathbb M_4(K)$, $x\mapsto E_{31}+E_{32}$, $y\mapsto E_{41}+E_{42}$, in terms of the elementary matrices $E_{ij}$, and the vectors $l_1=e_1$ and $l_2=e_2$ in $L$.

Then $Ae_1$ is the three dimensional subspace spanned by $e_1,e_3,e_4$, and $Ae_2$ is the subspace spanned by $e_2,e_3,e_4$, so their intersection $M$ is the two dimensional subspace spanned by $e_3,e_4$. Now $x,y$ act trivially on $M$, so $M\cong S^2$ is isomorphic to two copies of the trivial module $S=A/(x,y)$. There is however no epimorphism $A\twoheadrightarrow S^2$, since every two dimenisonal quotient of $A$ is isomorphic to $K[t]/(t^2)$.