# Generalization of semi-hereditarity

Let $$R$$ be a ring. A left $$R$$-module $$K$$ is called an $$N$$-th kernel if there are projective left $$R$$-modules $$P_1, \ldots P_N$$ and a short exact sequence $$0\rightarrow K \rightarrow P_N \rightarrow \ldots \rightarrow P_1\,.$$

The last arrow need not be surjective. A $$0$$-th kernel is an arbitrary module, a $$1$$-st kernel a submodule of a projective module, and so on. The ring $$R$$ has global dimension $$\le N$$ if and only if all $$N$$-th kernels are projective.

Having global dimension $$1$$ is called hereditary and $$R$$ is called semi-hereditary if all finitely generated submodules of projective modules, i. e. all finitely generated $$1$$-st kernels, are projective.

Question: Is there a name for the property that all finitely generated $$N$$-th kernels are projective?