Quasi-projective modules over prime hereditary noetherian V-rings are projective or injective

Dinh, Hai Q.; Holston, Christopher J.; Huynh, Dinh Van

doi:10.1016/j.jalgebra.2012.04.002

Cited by 8 publications

(5 citation statements)

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“…However, we were able to show that if, in addition, the ring R is a right V-ring, then a right R-module P is projective if and only if P is R-projective (see our recent work [8]). Hence the V-condition makes prime right hereditary right Noetherian rings behave so nicely that they become ''test-rings'' for the projectivity in the category of their modules (just like all rings do for injectivity on their modules).…”

Section: Introductionmentioning

confidence: 91%

Some results on V-rings and weakly V-rings

Dinh¹,

Holston²,

Huynh³

2013

Journal of Pure and Applied Algebra

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a b s t r a c tA ring R is called a right weakly V-ring (briefly, a right WV-ring) if every simple right R-module is X -injective, where X is any cyclic right R-module with X R R R . In this note, we study the structure of right WV-rings R and show that, if R is not a right V-ring, then R has exactly three distinct ideals, 0 ⊂ J ⊂ R, where J is a nilpotent minimal right ideal of R such that R/J is a simple right V-domain. In this case, if we assume additionally that R J is finitely generated, then R is left Artinian and right uniserial with composition length 2. We also show that a strictly right WV-ring with Jacobson radical J is a Frobenius local ring if and only if the injective hull of J R is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.

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Section: Introductionmentioning

confidence: 91%

Some results on V-rings and weakly V-rings

Dinh¹,

Holston²,

Huynh³

2013

Journal of Pure and Applied Algebra

Self Cite

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“…In [11,Lemma 1], the authors prove that over a right nonsingular right V -ring, maxprojective right R-modules are nonsingular. Regarding the converse of this fact, we have the following.…”

Section: Proofmentioning

confidence: 99%

max-projective modules

Alagöz,

Büyükaşik

2019

Preprint

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A right R-module M is called max-projective provided that each homomorphism f : M → R/I where I is any maximal right ideal, factors through the canonical projection π : R → R/I. We call a ring R right almost-QF (resp. right max-QF ) if every injective right R-module is R-projective (resp. max-projective). This paper attempts to understand the class of right almost-QF (resp. right max-QF ) rings. Among other results, we prove that a right Hereditary right Noetherian ring R is right almost-QF if and only if R is right max-QF if and only if R = S × T , where S is semisimple Artinian and T is right small. A right Hereditary ring is max-QF if and only if every injective simple right R-module is projective. Furthermore, a commutative Noetherian ring R is almost-QF if and only if R is max-QF if and only if R = A × B, where A is QF and B is a small ring.

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“…In [14], C. Faith asked when R-projectivity implies projectivity for all right R-modules. This problem has been considered by several authors, see [25], [18], [21], [8], [2], [31], [32]. DOI 10.14712/1213DOI 10.14712/ -7243.2021 Characterizing rings by projectivity of some classes of their modules is a classical problem in ring and module theory.…”

Section: Introductionmentioning

confidence: 99%