An Artinian version of the Artin-Rees lemma and duality

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.