THE RING WHOSE MODULE CLASS IS EMBEDDED IN PROJECTIVE MODULE
Corresponding Author(s) : Banh Duc Dung
UED Journal of Social Sciences, Humanities and Education,
Vol. 2 No. 2 (2012): UED JOURNAL OF SOCIAL SCIENCES, HUMANITIES AND EDUCATION
Faith and Walker (1967) characterized QF rings as the class of rings if and if every right injective module is projective. It can be implied that if every right R-module is embedded in a projective module or, equivalently, in a free module, then R is QF. The question is if not all the module but a class of it is embedded, how the ring R is. The ring of which every finitely generated (cyclic) right R-module is embedded in a free module is called right FGF (resp. CF). There have been two conjectures:
Right FGF ring is QF (FGF’s conjecture)?
Right CF ring is artinian (CF’s conjecture)?
If the CF’s conjecture is true, then so is FGF’s conjecture because a right artinian and FGF ring are QF. In this paper, we introduce these problems generally and then pose some open questions.
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