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Let $A$ be an abelian group and let $R$ be a ring. Is it true that there exists a bijective correspondence between the left $R$-module structures on $A$ and the right $R$-module structures on $A$?


I think this should be true, based on the fact that a left $R$-module is a right $R^{\operatorname{op}}$-module. But this alone is not enough to prove the statement, it only changes the problem from finding a bijection between left and right $R$-module structures to finding a bijection between right and left $R^{\operatorname{op}}$-module structures. Also, I have found from here and here that $\begin{pmatrix}\mathbb{Q}&\mathbb{Q}\\0&0\end{pmatrix}$ has a different number of left and right ideals, but this doesn't seem enough to prove that the statement is false.
In conclusion, I'm confused. Any help is appreciated.

commie trivial
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  • Left $R$-module structures on $A$ are in bijection with ring homomorphisms $R\to End(A)$ and right $R$-module structures on $A$ are in bijection with ring homomorphisms $R^{op}\to End(A)$, or, equivalently, $R\to End(A)^{op}$. Thus it suffices to provide an isomorphism $End(A)\to End(A)^{op}$. – Kenta S Oct 16 '22 at 03:54
  • I believe this fails for something like $A=\mathbb Z\times\mathbb Z/2$, where $End(A)=\begin{pmatrix}\mathbb Z&0\\mathbb Z/2&\mathbb Z\end{pmatrix}$ – Kenta S Oct 16 '22 at 04:00
  • Thank you, this seems useful. Could you help me prove how is $\operatorname{End}(A)$ not isomorphic to its opposite ring? I don't understand how the multiplication works here. How can we multiply an element of $\mathbb{Z}$ with an element of $\mathbb{Z}/2$ when we perform matrix multiplication? – commie trivial Oct 17 '22 at 12:50

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