What do we call the analogous version of free abelian group when $A$ is an arbitrary abelian group?

Question

For homology with coefficients, we often have that the chain group $C_n(K)$ is comprised of elements of the form $\sum g_i\sigma_i$ where $g_i\in A$ for some abelian group $A$ and $\sigma_i$ is some $n$-simplex.

What do we call the structure for $C_n(K)$?

When $A=\mathbb{Z}$, we say that $C_n(K)$ is a free abelian group.

How about for other arbitrary $A$? Is there any name like "free something"?

Thanks.