Hatcher's Algebraic Topology defines singular and cellular homology taking coefficients in a general abelian group, rather than just $\mathbb{Z}$. However, all of the actual examples seem to use abelian groups that are naturally viewed as rings, namely fields and $\mathbb{Z}/n\mathbb{Z}$. This seems fairly natural, since then in this case the chain groups and homology groups have a module structure. That said, are there examples of situations where it is useful to consider homology over an abelian group that is not "naturally" a ring? (That is, an abelian group where it is not standard to endow it with a ring structure, regardless of if such a structure is theoretically possible.)
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2Homology with coefficients in direct sums (like vector spaces) are common, e.g. in spectral sequences. – Cheerful Parsnip Jan 14 '24 at 19:54