Sheaves and Algebraic Geometry

There is a strange definition of $\Omega^1_{B/A}$, the module of relative differentials of $B$ over $A$, which starts as follows:

Let $\mu\colon B\otimes_A B\to B$ be the multiplication map (induced by $B\times B\to B,\; (b,b')\mapsto bb'$, which is $A$-bilinear). View $B\otimes_A B$ as a $B$-module by left multiplication, and let $I$ be the kernel $\ker\mu$, viewed as a $B$-module. Then the $A$-linear map $d\colon B\to I,\; b\mapsto 1\otimes b - b\otimes 1$ induces an $A$-linear derivation $B\to I/I^2$.

It turns out that $I$ is generated $B$-linearly by the elements $db$, and in fact $\Omega^1_{B/A}$ can be realized (defined) as $I/I^2$ with universal $A$-linear derivation $B\to I/I^2$ given by $d$ as above.