I am having a little trouble understanding the following construction. Suppose $R$ is a commutative ring. Let $B$ be a unital associative $R$-algebra, and let $\{b_{i}\}_{i \in I}$ be a fixed generating set for $B$. If $M$ is a $B$-module, then the action of each $b_i$ defines an $R$-linear endomorphism $b_{i}^{M}$.
What is are these maps $b_{i}^{M}$? Is it just $b_{i}^{M}(m) = b_{i}m$ for all $m \in M$?
Yes, given no other information, that is the most likely meaning for what you are looking at. Notice how centrality of $R$ is necessary for proving $b_i^m(rm)=rb_i^M(m)$.