Singular Homology - Coefficients in R

Coefficients in R

Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module. That is, rather than performing the above constructions from the starting point of free abelian groups, one instead uses free R-modules in their place. All of the constructions go through with little or no change. The result of this is

which is now an R-module. Of course, it is usually not a free module. The usual homology group is regained by noting that

when one takes the ring to be the ring of integers. The notation H_n(X, R) should not be confused with the nearly identical notation H_n(X, A), which denotes the relative homology (below).