There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology groups of $G$ can be interpreted as those of $X$. Similarly there is a deep connection between algebraic topology of Lie groups and homological algebra on Lie algebras. So a natural question is: Is there any deep connection between algebraic topology and homological algebra on rings?
EDIT I mean by "homological algebra on rings" homological algebra over the abelian categories of modules over rings.