Annihilator of various generators of a module

Question

Is is possible for $m,n$ to be two generators of a module $M$ over a ring $R$ such that $$\operatorname{Ann}_R(m) \not \cong \operatorname{Ann}_R(n)\,\,?$$

We know that $R/\operatorname{Ann}_R(m) \cong R/\operatorname{Ann}_R(n)$.

It's not possible in the commutative case, since then they are both equal to $\operatorname{Ann}_RM$.

Truth be told, I don't even know an example of two left ideals $L, L' \triangleleft R$ where $L \not \cong L'$ but $$R/L \cong R/L'$$ as left $R$ modules.

Answer 1

Hint: Let $k$ denote any field and $R:=\mathbb M_n(k)$ denote the ring of $n\times n$ matrices over $k$. What are the maximal left ideals $I$'s of $R$? What does $R/I$ look like?