Can anyone give an explicit basis of the universal (noncommutative) differential calculus over an algebra $A$ with basis ${e\_i}$. (The universal calculus over $A$ is the kernel of the multiplication map $m:A \otimes A \to A$.)



Basis for Universal Calculus
You can describe $\Omega\_A=\ker(m:A\otimes A\to A)$ as the quotient of the free $A$-bimodule generated by symbols $d(a)$, one for each element $a\in A$, by the sub-bimodule generated by the elements of the form $$d(ab)-d(a)\\,b-a\\,d(b), \qquad a,b\in A,$$ together with the elements of the form $$d(\lambda 1), \qquad \lambda\in k$$ with $k$ being the base field.
The elements $\{a d(b):a,b\in A\}$, when seen in $\Omega\_A$, span $\Omega\_A$ over $K$ but are not linearly independent over $k$.


To extract from this a $k$-*basis* of $\Omega\_A$ you need to know more than a basis of $A$. For example, if you know a presentation of $A$ given by generators and relations, you can obtain a basis using essentially Groebner bases.


