An Artinian version of the Artin-Rees lemma and duality

İsmail Hakkı Denizler; İsmail Hakkı Denizler

doi:10.3934/math.2026213

AIMS Mathematics

2026, Volume 11, Issue 2: 5219-5230. doi: 10.3934/math.2026213

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Research article

An Artinian version of the Artin-Rees lemma and duality

İsmail Hakkı Denizler ^,

Department of Mathematics, Faculty of Science, Yüzüncü Yıl University, 65080, Van, Turkey

Received: 21 November 2025 Revised: 22 January 2026 Accepted: 02 February 2026 Published: 28 February 2026
MSC : 13E10

We have presented a practical theorem for Artinian modules in this study, which is called the Artinian version of the Artin-Rees lemma. Using his own theorem [5, Theorem 1], D. Kirby demonstrated an Artinian analogue of the Artin-Rees lemma. Using the duality inferred from [2, Theorem 10.2.12]), we demonstrated how such an Artinian version may be directly and simply deduced from the original result [10, Theorem 8.5]. It can be generalized in a way that does not need the ring used to be local. We should point out that Matsumura ^[10] was not the first author to publish the Artin-Rees lemma. Note that Lang ^[7] published this theory earlier, in 1965. Also, Rees ^[13] published the ideal version of this theory in 1956.
- Noetherian modules,
- Artinian modules,
- duality
Citation: İsmail Hakkı Denizler. An Artinian version of the Artin-Rees lemma and duality[J]. AIMS Mathematics, 2026, 11(2): 5219-5230. doi: 10.3934/math.2026213

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Abstract

We have presented a practical theorem for Artinian modules in this study, which is called the Artinian version of the Artin-Rees lemma. Using his own theorem [5, Theorem 1], D. Kirby demonstrated an Artinian analogue of the Artin-Rees lemma. Using the duality inferred from [2, Theorem 10.2.12]), we demonstrated how such an Artinian version may be directly and simply deduced from the original result [10, Theorem 8.5]. It can be generalized in a way that does not need the ring used to be local. We should point out that Matsumura ^[10] was not the first author to publish the Artin-Rees lemma. Note that Lang ^[7] published this theory earlier, in 1965. Also, Rees ^[13] published the ideal version of this theory in 1956.

References

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