Module over a $K$ -monoid

Let $𝐴$ be a $K$ -monoid over $𝕂$ . A (left) $𝐴$ -module is a $K$ -module $𝑉$ equipped with a bilinear map

𝐴 \times 𝑉 \to 𝑉 (𝑎, 𝑣) \mapsto 𝑎 \cdot 𝑣

such that

$1 \cdot 𝑣 = 𝑣$ for $𝑣 \in 𝑉$
$(𝑎 𝑏) \cdot 𝑣 = 𝑎 \cdot (𝑏 \cdot 𝑣)$ for $𝑎, 𝑏 \in 𝐴$ , $𝑣 \in 𝑉$

which is a curried version of a unital algebra homomorphism

𝐴 \to E n d K (𝑉) .

We also call this a representation of $𝐴$ carried by $𝑉$ .

Properties and further terminology

$𝑉$ automatically carries a Lie algebra representation of the commutator algebra of $𝐴$ and any Lie subalgebra.
A submodule of $𝑉$ is an invariant subspace under the action of $𝐴$ .
A module is irreducible iff it has no proper nontrivial submodules.
A module is indecomposable iff it cannot be decomposed into the direct sum of two nonzero submodules.
The regular representation shows that $𝐴$ is a module over itself.

Explanation

Since a $K$ -monoid $𝐴$ is itself a ring, it is possible to form a module $𝑉$ over $𝐴$ . The action of $𝕂$ on $𝐴$ and $𝐴$ on $𝑉$ induces an action of $K$ on $𝑉$ , thus the module $𝑉$ inherits the $K$ -linear structure of the underlying ring $𝐴$ . Therefore $𝑉$ is a $K$ -module.