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Title:
A note on Noetherian and Artinian hoops
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Discussiones Mathematicae - General Algebra and Applications
Received: 2021-07-19 , Revised: 2023-01-17 , Accepted: 2023-01-17 , Available online: 2023-06-05 , https://doi.org/10.7151/dmgaa.1435
Abstract:
The aim of this paper is defining the concepts of Noetherian and Artinian hoops
by using the filter of hoop in the partial order set of all the filters of hoops
and inclusion relation and find some equivalent definitions for this notion.
We translate some important results from theory of rings to the case of hoop
and their characterizations are established. The relation between short exact
sequence on Noetherian and Artinian hoop studied and by using short exact
sequence we prove that the Cartesian product of two hoops is Noetherian
(Artinian) if and only if each one is a Noetherian (Artinian). By using the
notion of filter in hoops, we define the notion of composition series and prove
any $\vee$-hoop is Noetherian and Artinian if and only if it has composition
series. Finally, Chinese Remainder theorem in hoop and the relation between
maximal filter and Noetherian (Artinian) hoop are investigated.
Keywords:
hoop, Noetherian hoop, Artinian hoop, filter, Chinese reminder, composition series
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