$k$-linear derivations of a (not necessarily commutative) associative unital $k$-algebra$A$ with values in $A$-bimodules are often considered: these are $k$-linear maps $d: A\to M$ satisfying the Leibniz identity: $d(a b)=d(a) b+ a d(b)$. One would like to construct a universal construction of that kind.

Definitions

The triple $(A,M,d)$ is then called a differential calculus over $A$, provided $M$ is in the $k$-linear span of all expressions of the form $a d(b_1) d(b_2)\cdots d(b_n)$ where $a,b_1,\ldots,b_n\in A$. Morphisms of differential calculi over $A$ are straightforward to define. A universal object in that category is the universal differential calculus and $\Omega A = M$ in that case is called the universal differential envelope of $A$. It is constructed as the quotient of the tensor algebra $T A$ modulo the ideal generated by the (“augmentation”) kernel $I\subset A\otimes A$ of the multiplication map $A\otimes A\to A$; then $d a = 1\otimes a - a\otimes 1$.

This is a bit confusing. The subbimodule $I$ should rather be the degree-1 component of $\Omega A$ (see Bourbaki, Algèbre III, paragraph Problème universel pour les dérivations: cas non commutatif). —Benoit Jubin

Eric: I’m probably confused, but $A$ is degree-0 and $A\otimes A$ is degree-1.

Benoit Jubin: I agree. To me, we should have $\Omega A = A \oplus I \oplus \ldots$, whereas the article currently proposes $\Omega A = TA / \langle I \rangle$.

The bimodule $\Omega A$ is in fact a $k$-algebra, generated by degree-$1$ elements (in $A$); differential $d:A\to \Omega A$ can be naturally extended along the inclusion $A\hookrightarrow \Omega A$ to a differential $d:\Omega A\to \Omega A$ satisfying the Leibniz rule, by setting $d 1 = 0$, $d (w y) = d(w) y + (-1)^p w d(y)$, where $w,y$ are forms and $w\in\Omega^p A$ is homogeneous.