Hochschild Cohomology and Higher Order Extensions of Associative Algebras

The $n$th Hochschild cohomology group is described by $(n-2)$-extensions (Theorem 1). When $n=2,3$, the theorem reduces to the well-known classical results; for $n=1$, we get a description of the group of derivations by extensions; and for $n\ge 4$, this gives us a new description of cohomology groups. One can consider this theorem as an alternative definition of cohomology theory. So, one has some kind of hint to define cohomology theory for various algebraic structures.