OK, this question may be induced by my highly commutative mind and the wee midnight hours. Why do we bother with right R-modules vs left R-modules in view of the fact that given, say, a left R-module (M,+,$*$) (with scalar multiplication $*$) it is instantly a right R-module (M,+,$\circ$) with same addition and right multiplication $m\circ r =: r*m$ -- I believe the axioms for a right module check. In other words, in this way we have an isomorphism of the categories of right and left R-modules: RMod $\cong$ ModR ??? Thus, whatever we say about one of these categories (or their objects and morphisms) is instantly true for the other? No? I may be wrong, but do people use a phrase "let us assume that M is an R-R bimodule" or something like that? Could they mean M is a left R-module and a right R-module with some secret scalar multiplication on the right, other than the one I defined above?
Now that @Qiaochu et al found an error in my thinking, pointing out that the categories of right and left R-modules may be quite different, let alone isomorphic, then why is it, that one waives one's hands often saying "This argument works exactly the same way for right R-modules as the one we just exhibited for the left R-modules." ? This in spite of the fact that the two categories may be totally different. Do we have any fair mechanisms enabling us to know when the proofs are left-right independent, or is it that we can say that only after we actually check the proofs one line at a time...?