Is there a specific term for an abelian group (or ring) $G$ that satisfies the following property?
For every element $g \in G$ and natural number $n$, there exists a $q \in G$ such that $\underbrace{q + q + \dots + q}_{n\text{ times}} = g$.
Loosely speaking, this property means that you can "divide by natural numbers", even if you can't divide arbitrary (nonzero) elements as with a field. Obviously any field extension of the rational numbers satisfies this property, but I suspect that there are probably non-field examples as well.
The reason I ask is that it seems that me that the property above gives the minimum amount of algebraic structure necessary to define the arithmetic mean of elements of a set, which seems like kind of a natural thing to study.
This is called a divisible group.