What do you call this generalization of a module?

Question

A module $M$ is an Abelian group with the extra property that any element $m \in M$ can be multiplied by any element $r$ in a ring $R$. I want to relax this definition so that the $M$ need not be an Abelian group, but instead needs to only be a monoid. Does this structure have a standard name?

Answer 1

It's called a module. That is, $M$ must already be an abelian group. As Hagen observes in the comments, if $m \in M$ is an element then $(-1)m \in M$ is its additive inverse, so $M$ is a group. Now

$$(-1)(m + n) = (-1)m + (-1)n$$

by distributivity, but

$$(-1)(m + n) = (-1)n + (-1)m$$

in any group, from which it follows that $M$ is abelian. (In other words, a group is abelian if and only if taking inverses is a homomorphism.)

If $R$ is also relaxed so that it is a semiring, then $M$ can be a commutative monoid and then is referred to as a semimodule. I do not immediately see how to prove that $M$ must be commutative, but asking $M$ to be noncommutative is unnatural from the abstract point of view on what a semiring is, namely that a semiring is a thing that acts on commutative monoids. $M$ also has to be pretty close to commutative anyway, since similar to the above we have

$$2(m + n) = 2m + 2n = (m + n) + (m + n)$$

hence if $M$ is cancellative then it is commutative, and even if $M$ is not cancellative it has to be fairly close to be commutative even to be, say, a $\mathbb{Z}_{\ge 0}$-semimodule.