Non commutative truncated polynomial extensions

We introduce the notion of non commutative truncated polynomial extension of an algebra A. We study two families of these extensions. For the first one we obtain a complete classification and for the second one, which we call upper triangular, we find that the obstructions to inductively construct them, lie in the Hochschild homology of A, with coefficients in a suitable A-bimodule.Comment: 29 pages, We add some results about simplicirty (Prop. 1.11 and 3.6) and a result about rigidity (Th. 4.17