Let $R$ be a commutative ring.
Then, we say $M$ is an $R$-module instead of left $R$-module or right $R$-module or $(R,R)$-bimodule.
I'm curious why this convention is acceptable in general.
Let's say $M$ is an $R$-module as the usual convention.
Then, we don't know whether $M$ is indeed a left or right $R$-module. However, if $M$ is given one of left or right module structure, then we can convert $M$ to an $(R,R)$-bimodule such that $rx=xr$. So, in this point of view, it seems reasonable to call it just an $R$-module.
However, consider $\mathbb{C}\otimes_{\mathbb{R}} \mathbb{C}$ equipped with the natural $(\mathbb{C},\mathbb{C})$-bimodule structure.
Even though this is a module over a commutative ring and it is a bimodule, its left and right action do not coincide at all ($i•(1\otimes 1)\neq (1\otimes 1)•i$).
So when I write an argument I always clarify whether a module is right or left or symmetric or not (i.e. $xr=rx$).
Why does people use this convention? Or am I misunderstanding something? That is, does "$R$-module" actually mean a symmetric $(R,R)$-bimodule? If so, this term is acceptable to me, but I have seen many articles using this term to mean more than that.
Yes: when $R$ is a commutative ring, "$R$-module" means "symmetric $\left(R,R\right)$-bimodule". These are identified with left $R$-modules and with right $R$-modules as you would expect (but not with arbitrary $\left(R,R\right)$-bimodules).