Singular homology with coefficients in a ring versus in an abelian group

Solution 1:

If $R$ is a ring, it has an underlying abelian group : just forget the multiplication and the $1$. It is common to denote this underlying abelian group by the same symbol (here $R$) and use phrases such as "we view $R$ as an abelian group".

In particular, singular homology with coefficients in a ring is a special case of singular homology with coefficients in an abelian group.

However, the extra structure on $R$ affords extra structure on $H_n(X;R)$, namely that of an $R$-module . More generally, if $M$ is an $R$-module, then it also has an underlying abelian group and you can take $H_n(X;M)$ : this has the extra structure of an $R$-module too.

Overall, people tend to remove "forgetful" functors from the notation (so use the same letter for an $R$-module and its underlying abelian group, a ring and its underlying abelian group, etc.), at least the common ones.