Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to the relations
$$ d(\lambda a + \mu b) = \lambda da + \mu db, ~~~~~~ d(ab) = a\,db + da \,b$$
for $a, b \in A$, $\lambda, \mu \in k$. Higher degree forms are then defined by
$$\Omega^n_A = \Omega^1_A \otimes_A \cdots \otimes_A \Omega^1_A.$$
The direct sum of all forms $\Omega_A = \sum_{n=0}^\infty \Omega_A^n$ becomes a differential graded algebra with the obvious differential. 


Now Loday's book "cyclic cohomology", he defines the non-commutative de-Rham cohomology of $A$ to be the cohomology of the complex $((\Omega_A)_{\mathrm{ab}}, d)$, i.e. he takes the abelianization of $\Omega_A$. 

**Q:** Why is that? What is wrong with defining the non-commutative de-Rham-cohomology just as the cohomology of $\Omega_A$?