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?



Name for the action of a bialgebra on an algebra
According to nLab, such an action is called a [Hopf action](https://ncatlab.org/nlab/show/Hopf+action) and your data specify a [left $B$-module algebra](https://ncatlab.org/nlab/show/module+algebra). Such a structure is also referred to in the literature as an [algebra in the category](https://mathoverflow.net/q/246881/85967) (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](https://en.wikipedia.org/wiki/Group_Hopf_algebra#Hopf_module_algebras_and_the_Hopf_smash_product) or a Hopf-module algebra or an [algebra in the category](https://mathoverflow.net/q/246881/85967) (of left $H$-modules).  

(You can also check section 3.1 of [this article](http://www.mdpi.com/2075-1680/1/1/74), 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](https://ncatlab.org/nlab/show/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](http://www.ams.org/journals/tran/1986-298-02/S0002-9947-1986-0860387-X/S0002-9947-1986-0860387-X.pdf) 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. 


