• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.141-154
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• 2022

Let R be a commutative ring with identity. Denote by w𝒫_w the class of weak w-projective R-modules and by w𝒫_w^⊥ the right orthogonal complement of w𝒫_w. It is shown that (w𝒫_w, w𝒫_w^⊥) 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.