Corollary

Let $S$ be a generating system of the algebra $A$. In order that $d' \circ \rho = \theta \circ d$, it is necessary and sufficient that $d'(\rho(x)) = \theta(d(x))$ for all $x \in S$.

This is an immediate consequence of Proposition 9 and no. 5, Corollary to Proposition 3.

Under the conditions of Proposition 9, we know that $B$ has (by means of $\rho$) an $(A, A)$-bimodule structure (II, § 1, no. 14, Example 1).