Existence of nonprojective quasi-Gorenstein projective modules

Determine whether there exists an associative ring R for which the class of quasi-Gorenstein projective left R-modules is strictly larger than the class of projective left R-modules, that is, whether \(\mathcal{QGP}(R)\ne\operatorname{Proj}R\).

Background

The paper develops quasi-Gorenstein projective modules and proves that $\mathcal{QGP}(R)=\operatorname{Proj}R$ for several important classes of rings, including left perfect rings, commutative Noetherian rings of finite Krull dimension, and rings satisfying the finite test-module hypotheses established in Theorem 5.7. These results leave unresolved whether the equality holds for every associative ring.

Proposition 5.1 shows that if a nonprojective module $G$ belongs to $\mathcal{QGP}(R)$, then its quasi-projective dimension is infinite. In the commutative case, Corollary 5.4 additionally shows that such a module cannot be finitely generated. Thus the problem asks whether genuinely nonprojective quasi-Gorenstein projective modules can occur, necessarily outside the classes and finiteness conditions settled in the paper.