Alagoz, YusufMoradzadeh-Dehkordi, Ali2024-12-242024-12-2420242651-477Xhttps://doi.org/10.15672/hujms.1186239https://hdl.handle.net/20.500.12604/7466In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring R is two-sided K & ouml;the if all right R-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with J(R)(2) = 0, it is proven that all R-modules are min-pure projective if and only if R is either a field or a quasi-Frobenius ring of composition length 2.eninfo:eu-repo/semantics/openAccess(min-)purityK & ouml;the ringsuniversally mininjective ringsquasi-Frobenius ringsArticle532342355N/AWOS:001236037800001Q22-s2.0-8519298110510.15672/hujms.1186239