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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Definition of Non-commutative de-Rham-Cohomology
I do not really know the reason, but if I would guess I think it might be $\Omega(A)$ is no longer an abelian category. The exact sequences you written down are not just exact sequences of abelian gro …