In what follows, the coefficient group $A$ is sometimes not written. It is common to take $A$ to be a commutative ring $R$; then the cohomology groups are $R$-modules. A standard choice is the ring $\mathbb Z$ of integers.
My first confusion is: if $A$ is supposed to be an abelian group, how can one instantiate $A$ with a commutative ring $R$? A commutative ring is a completely different type of object than an abelian group.
Question: How does singular homology with coefficients in a ring relate to singular homology with coefficients in an abelian group? Is one of these concepts more general?