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?
