Submodules and direct sum

Question

This question looks silly but I got little bit confused. Let $R$ be an integral domain and let $A, B, C$ and $D$ be the $R$-modules. Let $A \bigoplus B = C \bigoplus D,$ where all $A, B, C$ and $D$ cannot be expressed as a direct sum of $R$-modules. Suppose $A$ and $B$ are $R$-module embedding into $C$ and $D$ respectively. Is that will be enough to say $A = C$ and $B = D$? I am little bit worried about here $A$ and $B$ are not the submodules of $C$ and $D$ respectively but identify as $R$-submodules by embedding.