The Dual of the Notion of “Finitely Generated”

Vámos, Peter

doi:10.1112/jlms/s1-43.1.643

Cited by 92 publications

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“…Following J. P. Jans [16], a ring R is called right co-Noetherian if every factor module of every finite cogenerated right R-module is again finitely cogenerated or, equivalently, every finitely cogenerated right /^-module is Artinian. As a consequence of results of P. Vamos [26], R is right co-Noetherian if and only if each simple right Rmodule has an Artinian injective hull. Thus any right co-Noetherian ring R satisfies property (*) of Theorem 5.…”

Section: Remarks and Examplesmentioning

confidence: 99%

On semi-artinian modules and injectivity conditions

Clark¹,

Smith²

1996

Proceedings of the Edinburgh Mathematical Society

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Section: Remarks and Examplesmentioning

confidence: 99%

On semi-artinian modules and injectivity conditions

Clark¹,

Smith²

1996

Proceedings of the Edinburgh Mathematical Society

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“…An It module M in said to be finitely embedded (f.e) [18] (later called by J.P. Jans [8] as co-fmitely generated) if, E(M] = E(S 1 )  E(S 2 )  .....  E(S n ) for some simple R modules S 1 ,...,S n (here E(A) denotes the injective hull of an R-module A).…”

Section: Definitionsmentioning

confidence: 99%

Empirical Study of Cocyclic Copurity and the Dualization of Cyclic Purity

Arshaduzzaman¹,

Perwej²,

Kumar³

2016

IJAIS

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“…A right R -module is finitely embedded if it has finitely generated essential socle (see, e.g., Vamos [4]). In [1] Jategaonkar settled the Jacobson conjecture for left and right fully bounded noetherian rings by showing that every finitely generated finitely embedded module is artinian.…”

mentioning

confidence: 99%

Finitely embedded modules over Noetherian rings

Ginn¹,

Moss²

1975

Bull. Amer. Math. Soc.

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A right R -module is finitely embedded if it has finitely generated essential socle (see, e.g., Vamos [4]). In [1] Jategaonkar settled the Jacobson conjecture for left and right fully bounded noetherian rings by showing that every finitely generated finitely embedded module is artinian. It is an open question whether or not this property holds for an arbitrary left and right noetherian ring (though it is well known that right noetherian is not enough). We prove here that if R is left and right noetherian and M is a projective finitely generated finitely embedded module over R, then M is artinian. This result can be extended to cover the case where M is an arbitrary finitely embedded submodule of a finitely generated free R-module. The proof of this and related results will appear elsewhere. Even the case M = R seems to be new and in this case we can obtain the more general THEOREM. If R is a left and right noetherian ring and if the right socle of R is either left or right essential, then R is artinian.We note that there exist right noetherian rings with right essential right socle which are not right artinian (see, e.g., [2] ).All rings have identity and modules are unitary. l A (X) and r A (X) denote, respectively, the left and right annihilators of Xin the ring 4.We require the following result of T. H. Lenagan [3] which is adapted to suit our purpose in the form of We also need the following lemma whose proof is straightforward.

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The Dual of the Notion of “Finitely Generated”

Cited by 92 publications

References 0 publications

On semi-artinian modules and injectivity conditions

On semi-artinian modules and injectivity conditions

Empirical Study of Cocyclic Copurity and the Dualization of Cyclic Purity

Finitely embedded modules over Noetherian rings

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