Let A be a Banach algebra and let X be a Banach
A-bimodule. In studying H¹(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