Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that $$ (b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a). $$ When it also holds that $$ b \triangleright(ac) = (b_{(1)} \triangleright a)(b_{(2)}\triangleright c) $$ what do we call this kind of action?

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According to nLab, such an action is called a Hopf action and your data specify a left $B$-module algebra. Such a structure is also referred to in the literature as an algebra in the category (of left $B$-modules).

Note also, that, if instead of the bialgebra $B$ we consider a hopf algebra $H$ acting on $A$ and satisfying your condition supplemented by: $$ h\triangleright 1_A=\varepsilon(h)1_A $$ where $\varepsilon:C\rightarrow k$ is the counity map of $H$, then under such an action, $A$ is called a $H$-module algebra or a Hopf-module algebra or an algebra in the category (of left $H$-modules).
(You can also check section 3.1 of this article, for a review on this and other similar kinds of actions and coactions on algebras and coalgebras).

Edit: If we drop the demand for the bilinear map $\triangleright: H \times A \to A$ to be an action, i.e., if we drop $(h_1h_2) \triangleright a = h_1\triangleright(h_2 \triangleright a)$ and simply require $h \triangleright(ac) = (h_{1} \triangleright a)(h_{2}\triangleright c)$ and $h\triangleright 1_A=\varepsilon(h)1_A$, then we say that the bilinear map $\triangleright$ is a measuring or that $H$ measures $A$. If, furthermore: $1_H\triangleright a=a$ for all $a\in A$ then we are speaking about a weak action of $H$ on $A$ (see def.1.1, p.674 of the linked article). The same terminology is used for the bialgebra case as well.

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