Do we have $R\simeq S$ for two submodules $R,S$ of $A^n$? 
Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n$ (where $n\in\Bbb N$), if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, then do we have $R\simeq S$?

Note that this is definitely false for quotients of non-free modules: see, e.g., Quotient modules isomorphic $ \Rightarrow$ submodules isomorphic or Isomorphy of quotient modules implies isomorphy of submodules .

No.
Let $A=\Bbb Z^\Bbb N=\{\,f\colon \Bbb N\to\Bbb Z\,\}$, $n=1$, $R=\Bbb Z=\{\,f\in A\mid \forall n>0\colon f(n)=0\,\}$, and $S=0$. Then $A^1/R\cong A^1/S\cong A$, but of coure $R\not\cong S$. 
