Let C n(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B....

An $𝒮$-closed submodule of a module $M$ is a submodule $N$ for which $M/N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$-closed submodule $N$ of $M$ is a direct summand of $M$. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$-modules are projective if and only if all right $R$-modules are GCS-modules.