We introduce the notion of crossed $n$-fold extensions of an algebra $B$ by a bimodule $M$ and prove that such extensions represent classes in the Hochschild cohomology of $B$ with coefficients in $M$. Moreover we consider this way characteristic classes of chain (resp. cochain) algebras in Hochschild cohomology.
"Crossed extensions of algebras and Hochschild cohomology." Homology Homotopy Appl. 4 (2) 63 - 82, 2002.