  Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic
study of commutative unitary rings equipped with a {\em biderivation}, namely a
binary operation that is a derivation in each argument, is here begun, with an
eye toward the geometry of the corresponding {\em $B$-varieties}. Foundational
results about extending biderivations to localisations, algebraic extensions
and transcendental extensions are established. Resolving a deficiency in
Poisson algebraic geometry, a theory of base extension is achieved, and it is
shown that dominant $B$-morphisms admit generic $B$-fibres. A bidifferential
version of the Dixmier-Moeglin equivalence problem is articulated.
