Test sets for factorization properties of modules

Abstract

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property with respect to\ a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring $R$ and on additional set-theoretic hypotheses. For $R$ commutative noetherian of Krull dimension $0<d<\infty$, we show that the assertion 'projectivity is a factorization property with respect to a single epimorphism' is independent of ZFC + GCH. We also show that if $R$ is any ring and there exists a strongly compact cardinal $\kappa >|R|$, then the category of all projective modules is $\kappa$-accessible.