  Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the
bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a
given derivation D: A-> X to a Banach algebra B containing A as an ideal,
thereby exploiting (or establishing) hereditary properties. This is usually
done using (bounded/unbounded) approximate identities to obtain the extension
as a limit of operators b->D(ba)-b.D(a), (a in A) in an appropriate operator
topology, the main point in the proof being to show that the limit map is in
fact a derivation. In this paper we make clear which part of this approach is
analytic and which algebraic by presenting an algebraic scheme that gives
derivations in all situations at the cost of enlarging the module. We use our
construction to give improvements and shorter proofs of some results from the
literature and to give a necessary and sufficient condition that biprojectivity
and biflatness are inherited to ideals.
