What is the correct term for a module but with the scalars being elements of a monoid?

Question

From my understanding, a module is similar to a vector space over a field, except the scalars come from a ring. What about the cases when the scalars come from a monoid? Is there a specific term used when the scalars are from a monoid instead?

Answer 1

Some people call them monoid actions, by analogy with group actions. There's also the term "monoid act," but I don't know who uses it.

Answer 2

We call them modules over the monoid. In general, for a monoid object $M$ in a monoidal category $\mathbf{C}$, we call objects of $\mathbf{C}$ on which $M$ acts $M$-modules. These form a category. See this for more information.