$n$-Copure Projective Modules

Let $R$ be a ring, $n$ a fixed nonnegative integer and $\mathcal{F}_n$ the class of all left $R$-modules of flat dimension at most $n$. A left $R$-module $M$ is called $n$-copure projective if $\operatorname{Ext}_R^1(M,F)=0$ for any $F\in \mathcal{F}_n$. Some examples are given to show that $n$-copure projective modules need not be $m$-copure projective whenever $m>n$. Then we characterize the well-known QF rings and IF rings in terms of $n$-copure projective modules. Finally, we prove that a ring $R$ is relative left hereditary if and only if every submodule of a projective (or free) left $R$-module is $n$-copure projective if and only if $\operatorname{id}_R(N)\leqslant 1$ for every left $R$-module $N$ with $N\in \mathcal{F}_n$.