Modules I: Basic definitions and constructions

In constrast to group actions, however, one cannot always convert left module structures to right structures and vice versa. To clarify the situation, it is convenient to introduce the opposite ring R, which has the same underlying abelian group R but with multiplication a · b = ba. Then a left module over R is the same thing as a right module over R, and vice versa. Consequently, if R happens to be isomorphic to R—preferably in some canonical way—then we can convert back and forth between left and right actions in much the same way as we did for group actions. Explicitly, suppose we are given an isomorphism α : R ∼= −→ R, and M is a right R-module. Then r ? x = xα(r) (r ∈ R, x ∈ M) makes M into a left R-module. Note also that such an isomorphism α is the same thing as an anti-automorphism of R, i.e. an isomorphism of abelian groups α : R ∼= −→ R such that α(ab) = α(b)α(a) for all a, b ∈ R. Many familiar rings admit such anti-automorphisms: