Projectivity classes, which dualize injectivity classes (cf.[ll]), are introduced and some examples are given. Characterizations of hereditary rings, semisimple rings, Noetherian rings, (semi-)perfect rings, quasi-perfect rings, semiregular rings, semiregular modules and F- semiperfect modules using projectivity classes are given. Finally, for a projectivity class P, P-projective covers are defined and similar results with “quasi-projective cover” substituted by “T-projective cover” will still hold. Our results unify and generalize several well known results by Golan, Huynh and Smith, Rangaswamy and Vanaja, Tiwary and Pandeya, and Xue.
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