Derivation of an algebra

Let $A\stackrel{\phi }{\to }B$ be a homomorphism of commutative rings (Commutative ring), so that $B$ is an $A$-algebra, and let $M$ be a $B$-module (Unital left and right modules over a ring; unqualified module means left module). An $A$-derivation of $B$ into $M$ is a map $D\text{\hspace{0.17em}⁣}:B\to M$ satisfying, for all $b,b^{\mathrm{\prime }}\in B$ and all $a\in A$, the three laws

$D(b+b^{\mathrm{\prime }})=D(b)+D(b^{\mathrm{\prime }}),D(\phi (a))=0,D(b{b}^{\mathrm{\prime }})=bD(b^{\mathrm{\prime }})+b^{\mathrm{\prime }}D(b)\mathrm{.}$

The first law says that $D$ is additive; the second that $D$ is $A$-constant (it kills the image of $A$); the third is the Leibniz rule. The set of all such maps is written ${\mathrm{Der}}_A(B,M)$. It is a $B$-module under the pointwise operations $(D+D^{\mathrm{\prime }})(b):=D(b)+D^{\mathrm{\prime }}(b)$ and $(c\cdot D)(b):=cD(b)$: the sum and scalar multiples are again additive $A$-constant maps satisfying Leibniz, because each law is linear in $D$, and the zero map is a derivation.

Three conventions are part of the definition.

1. No finiteness. Nothing is assumed about $B$ as an $A$-algebra: it need not be finitely generated, finitely presented, or flat, and $A$ need not be Noetherian. The definitions used later on this page are the same ones used for the earlier algebraic-differentials interface of this track.
2. $A$-linearity, not $B$-linearity. Every $A$-derivation is $A$-linear in the sense that $D(\phi (a)b)=\phi (a)D(b)$ for $a\in A$, $b\in B$: by Leibniz, $D(\phi (a)b)=\phi (a)D(b)+bD(\phi (a))$ and the second term vanishes. A derivation is in general not $B$-linear, and this failure is exactly what the Leibniz rule measures; it also shows $D(1)=0$, since $1=\phi (1_A)$ makes $1$ $A$-constant.
3. Functored variables. For a fixed ring map $A\to B$ and a $B$-linear map $h\text{\hspace{0.17em}⁣}:M\to N$ of $B$-modules, composition $D\mapsto h\circ D$ is a $B$-module map ${\mathrm{Der}}_A(B,M)\to {\mathrm{Der}}_A(B,N)$. Consequently ${\mathrm{Der}}_A(B,-)$ is a functor from $B$-modules to $B$-modules, and the Leibniz rule is preserved by postcomposition with any module map.