Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring with identity. Denote by $\wp$ the class of weak $w$-projective $R$-modules and by $\wp^{\bot}$ the right orthogonal complement of $\wp$. It is shown that $(\wp,\wp^{\bot})$ is a hereditary and complete cotorsion theory, and so every $R$-module has a special weak $w$-projective precover. We also give some necessary and sufficient conditions for weak $w$-projective modules to be $w$-projective. Finally it is shown that when we discuss the existence of a weak $w$-projective cover of a module, it is enough to consider the $w$-envelope of the module.

Keywords: Weak $w$-projective precover, $w$-operation (theory), cotorsion theory

MSC numbers: 13C10, 13D05, 13D07, 13D30

Supported by: The authors would like to express their sincere thanks for the referee for his/her careful reading and helpful comments. This research was supported by the Academic Research Fund of Hoseo University in 2019 (20190817).