I am reading through Dummit and Foote and one of the examples I do not understand. The example is from 10.1, page 343.
(3) If $A$ is an $R$-algebra then the $R$-module structure of $A$ depends only on the subring $f(R)$ contained in the center of $A$.
How do you obtain an $R$-module structure from this?
An $R$ algebra is by definition a morphism of rings $f : R \to A$, where $A$ is a ring with $1$. For $a\in A$ and $r\in R$ one notes $ra = f(r)a$ and $ar = a f(r)$. Now this defines a structure of left and respectively right $R$-module on $A$. (Just try it.) Finally, $f(R)$ is in the center of $A$ if $R$ is a commutative ring, because then $bb' = b'b$ for all $b,b'\in f(R)$, as $R$ is commutative and $f$ a morphism of rings. Assuming directely that $f(R)$ is in the center does give you directly $bb' = b'b$ for all $b,b'\in f(R)$.