  We prove that for an inclusion of unital associative but not necessarily
commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild
homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in
Andr\'e-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is
flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an
H-unital kernel, one can express the Wodzicki excision sequence and the
corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic
(co)homology as a single long exact sequence.
