A characterization of noetherian rings by cyclic modules

It is 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. This provides an affirmative answer to a question raised by P. F. Smith.and Osofsky-Smith [20], respectively): Theorem A. 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 noetherian module.Theorem B. A ring R is right noetherian if every cyclic right R-module is a direct sum of a projective module and an injective module.

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“…We would like to remark further that these techniques are improvements of those that were developed in Huynh [14].…”

Section: Theorem C a Ring R Is Right Noetherian If And Only If Everymentioning

confidence: 99%

An Affirmative Answer to a Question on Noetherian Rings

Huynh

Rizvi

2008

J. Algebra Appl.

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Conditions for a ring to be Noetherian or Artinian

Plubtieng

Tansee

2002

Communications in Algebra

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Elementary preservation of the Noetherian condition and the potential applications in computational physics

Gomez-Ramirez

Vélez-Caidedo

Gallego-Gonzalez

2020

J. Phys.: Conf. Ser.

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We give an elementary proof of the preservation of the Noetherian condition for commutative rings with unity R having at least one finitely generated ideal I such that the quotient ring is again finitely generated, and R is I—adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. Furthermore, we discuss the potential applications that this new elementary proof possesses regarding the simplification of conceptual generations in mathematics and computational physics based on the new computational paradigm of Artificial Mathematical Intelligence. In addition, we give a counterexample showing that the ‘completion’ condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields.

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A characterization of noetherian rings by cyclic modules

Abstract: It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.1991 Mathematics subject classification: 16P40, 16D99.

Cited by 3 publications

References 20 publications

An Affirmative Answer to a Question on Noetherian Rings

An Affirmative Answer to a Question on Noetherian Rings

Conditions for a ring to be Noetherian or Artinian

Elementary preservation of the Noetherian condition and the potential applications in computational physics

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