What is the name of the algebraic structure constructed with an abelian monoid and a field? Take a vector space built from an abelian group $(V,+)$ whose elements are the vectors, a field $K$ whose elements are the scalars, and there is an operation (multiplication by scalars) that distributes on the group operation.
What is the name of the structure obtained if $(V,+)$ instead of being an abelian group is an abelian monoid?
Thanks and forgive my ignorance :)

There cannot be such a structure without the group information. If you use the same axioms replacing the group info with monoid info, you end up being able to show it's a vector space.
Proof: Let $v\in V$. Then as $1_F\cdot v=v$ we have that $v+(-1_F)v=(1_F-1_F)\cdot v=0\cdot v=0$. Because $V$ is closed under the field operation that $(-1_F)\cdot v$ is an additive inverse for $v$, hence $V$ is inherently an abelian group.
