TITLE: On Similarities and Differences Between Right and Left Modules over a Ring
QUESTION [17 upvotes]: I have read that for a ring $ R $ in general, right $ R $-modules are not the same things as left $ R $-modules.

Why do we say that a right $ R $-module is equivalent to a left $ R $-module only when $ R $ is commutative?
I feel that the commutativity of $ R $ is strictly an internal property of $ R $, so how can the commutativity of $ R $ affect scalar multiplication on an $ R $-module $ M $, which is an external operation?

REPLY [6 votes]: For a commutative ring $R$, whatever you can say (in module theoretic terms) about the category of left modules over $R$ can be said about the category of right modules, because there's a “canonical” way to turn any diagram of left modules into a diagram of right modules.
In the general case of noncommutative rings this is not true any more: rings can be left noetherian (artinian, coherent, perfect, …) and not right noetherian (artinian, coherent, perfect, …). All these properties can be defined in module theoretic terms via diagrams: a ring is left perfect when every left module has a projective cover, for instance; it is left noetherian when every direct sum of injective left modules is injective.
In case the ring $R$ has an involution, that is, a map $\varphi\colon R\to R$ such that

$\varphi^2$ is the identity,
$\varphi(r+s)=\varphi(r)+\varphi(s)$,
$\varphi(rs)=\varphi(s)\varphi(r)$,

then we are in a situation similar to the commutative case (where we can take $\varphi$ to be the identity): any left module $M$ can be turned into a right module with $xr=\varphi(r)x$ and it's easy to show that left module morphisms become right module morphisms. However, this is the exception rather than the rule for noncommutative rings.
Some module theoretic properties are left-right symmetric, though: semisimplicity and semiperfectness are examples.
The fact that every left module can be seen as a right module over the opposite ring simply tells that general properties of modules hold both for left modules and right modules: for instance, every module has an injective envelope, because a proof given for left modules doesn't depend on special properties of the ring.