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.
Best Answer
Ring theory is always lurking more or less visible in algebraic topology. As your question is quite broad, I'll try and give a quick overview with a few references. Since homological algebra takes place often over R-modules or it can be reduced to this case somehow, it might be a bit difficult to define what you mean by "homological algebra on rings". Nonetheless, here are a few examples that I think fit your description:
The Hochschild homology of an $R-R$-bimodule reflects some ring-theoretic stuff. For instance, $H_1(R,R)$ of a $k$-algebra $R$ is the module of differentials $\Omega_{R/k}$. If $\mathbb{Q}\subseteq R$ then there is an algebraic decomposition of this homology that is analogous to the Hodge decomposition in complex manifold theory.
Taking the above example further if $k$ is a ring and $X$ a simplicial set, the cyclic homology (Hochschild homology taking into account a cyclic action on the corresponding simplicial set) of the simplicial module $k[X]$ is the same as the $S^1$-equivariant homology of the geometric realisation of $X$ with coefficients in $k$.
(Hochschild and cyclic homology are related to $K$-theory and the Lie algebra of matrix algebras as well. For the above examples, see Chapter 9 of Weibel's book "An Introduction to Homological Algebra" and all of Loday's book "Cyclic Homology")
As mentioned in the comments, $K$-theory is like homology on rings. Moreover, algebraic topology is clearly interested in vector bundles; on a nice space $X$ the category of rank $n$ real vector bundles on $X$ is equivalent to the category of rank $n$ finitely generated projective modules over the continuous functions $C(X,\mathbb{R})$. The group $K_0(C(X))$ is the Grothendieck group of the isomorphism classes of fg projectives.
$K$-theory of course has its roots and many applications to algebraic topology and algebraic geometry. Plentiful examples can be found in "The Handbook of $K$-Theory" (http://www.math.uiuc.edu/K-theory/handbook/)
(In fact, the higher $K$-groups of a ring $R$ can be defined as the homotopy groups of a certain simplicial resolution associated to $R$; simplicial resolutions are like chain complexes in an abelian category, and in fact for abelian categories nonnegative chain complexes and simplicial sets coincide. So it really is like a homology of rings, as opposed to doing abelian homological algebra in the category of modules.)