Kähler differentials

Definition

Let $A\to B$ be an algebra. An A differential of B into an $A$-module $M$ is a map $D:B\to M$ such that

1. $D(a)=0$ for all $a\in A$
2. $D(b+b')=D(b)+D(b')$ for $b,b'\in B$
3. $D(bb')=b'D(b)+bD(b')$

Observe that the set of all such maps $Der_{A}(B,M)$ is a $B$-module. Moreover, $Der_{A}(B,-)$ is a representable functor; we call the representative $\Omega _{B/A}$ the module of Kähler differentials. That is, $\Omega _{B/A}$ satisfies the following universal property: