DOI: 10.15672/hujms.1186239 ISSN:

HOMOLOGICAL OBJECTS OF MIN-PURE EXACT SEQUENCES

Yusuf ALAGÖZ, Ali MORADZADEH-DEHKORDI
  • Geometry and Topology
  • Statistics and Probability
  • Algebra and Number Theory
  • Analysis
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In 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 Kothe 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.

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