An Affirmative Answer to a Question on Noetherian Rings

Huynh, Dinh Van; Rizvi, S. Tariq

doi:10.1142/s0219498808002631

Cited by 3 publications

(2 citation statements)

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“…Since then, there has been continuous work on finding properties on classes of modules that guarantee the ring to be right noetherian (or some other finiteness condition). For instance, if each cyclic right module is: an injective module or a projective module [5], a direct sum of an injective module and a projective module [12,14], or a direct sum of a projective module and a module Q, where Q is either injective or noetherian [6], then the ring is right noetherian. It is also known that if every finitely generated right module is CS, then the ring is right noetherian [7].…”

Section: Introductionmentioning

confidence: 99%

Rings Over Which Cyclics Are Direct Sums of Projective and Cs or Noetherian

Holston¹,

Jain

Leroy³

2010

Glasgow Math. J.

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Abstract. R is called a right WV-ring if each simple right R-module is injectiverelative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right Vring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. 'extending modules') or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.2000 Mathematics Subject Classification. 16D50, 16D70, 16D80.

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Section: Introductionmentioning

confidence: 99%

Rings Over Which Cyclics Are Direct Sums of Projective and Cs or Noetherian

Holston¹,

Jain

Leroy³

2010

Glasgow Math. J.

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“…In [24], it was shown that a ring R is right Noetherian if (and only if) every cyclic right R-module is a direct sum of a projective module and a module Q , where Q is either injective or Noetherian. But here, with our condition (℘), we cannot obtain the same result.…”

Section: When Are Wv-rings Noetherian?mentioning

confidence: 99%

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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