Representations of unital associative algebras

Let $k$ be a field, let $A$ be a unital associative $k$-algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), and let $V$ be a $k$-vector space. Write ${\mathrm{End}}_k(V)={\mathcal{L}}_k(V,V)$ for the $k$-linear endomorphisms of $V$ (The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication); its multiplication is composition and its identity is ${\mathrm{id}}_V$ (The endomorphism ring ${\mathrm{End}}_R(M)$ under addition and composition, Module endomorphisms form a ring under pointwise addition and composition). An (unital) representation of $A$ on $V$ is a $k$-linear map $\rho :A⟶{\mathrm{End}}_k(V)$ such that $\rho (1_A)={\mathrm{id}}_V$ and $\rho (ab)=\rho (a)\circ \rho (b)(a,b\in A)\mathrm{.}$ The associated action is $a\cdot v:=\rho (a)(v)$. The zero vector space is allowed; in that case ${\mathrm{id}}_V=0$.

This definition specifies an action of $A$ on one vector space. It does not by itself specify an action on $V{\otimes }_{k}W$. For the module interpretation and the extra coproduct data used for bialgebra modules, see Associative algebra representations are unital modules and The module category of a bialgebra and its tensor action.