universal derivation

Let $R$ be a commutative ring, and let $A$ be a commutative $R$-algebra. Auniversal derivation of $A$ over $R$ is defined to be an$A$-module ${\mathrm{\Omega }}_{A/R}$ together with an $R$-linear derivation $d:A\to {\mathrm{\Omega }}_{A/R}$, such that the following universal property holds:for every $A$-module $M$ and every $R$-linear derivation$\delta :A\to M$ there exists a unique $A$-linear map $f:{\mathrm{\Omega }}_{A/R}\to M$ such that $\delta =f\circ d$.

The universal property can be illustrated by a commutative diagram:

An $A$-module with this property can be constructed explicitly, so${\mathrm{\Omega }}_{A/R}$ always exists. It is generated as an $A$-module bythe set $\{dx:x\in A\}$, with the relations

$d(ax+by)$$=$$adx+bdy$

$d(xy)$$=$$x\cdot dy+ydx$

for all $a,b\in R$ and $x,y\in A$.

The universal property implies that ${\mathrm{\Omega }}_{A/R}$ is unique up toa unique isomorphism. The $A$-module ${\mathrm{\Omega }}_{A/R}$ is often calledthe module of Kähler differentials.