  Given a presentation of a monoid $M$, combined work of Pride and of Guba and
Sapir provides an exact sequence connecting the relation bimodule of the
presentation (in the sense of Ivanov) with the first homology of the Squier
complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This
exact sequence was used by Kobayashi and Otto to prove the equivalence of
Pride's finite homological type (FHT) property with the homological finiteness
condition bi-$\mathrm{FP}_3$. Guba and Sapir used this exact sequence to
describe the abelianization of a diagram group. We prove here a generalization
of this exact sequence of bimodules for presentations of associative algebras.
Our proof is more elementary than the original proof for the special case of
monoids.
