On Locally Artinian Rings

Adil Kadir Jabbar1

1 Department of Mathematics, Faculty of Science and Science Education, School of Science,  University of Sulaimani, Kurdistan Region, Iraq

Original: 6 July 2016, Revised: 17 September 2016, Accepted: 16 October 2016, Published online: 20 March 2017

DOI: https://doi.org/10.17656/jzs.10600

Abstract

In this paper, a new ring is introduced and studied, which we call a locally Artinian ring and it is a generalization of an Artinian ring. Several properties of Artinian rings are extended to this new type of commutative rings. Some conditions are given under which a locally Artinian ring is Artinian. It is known that, a locally Artinian ring is locally Noetherian, but the converse is not true and an example of a locally Noetherian ring which is not locally Artinian is given. Furthermore, a necessary and sufficient condition is given for a locally Noetherian ring to be locally Artinian.

Key Words: Artinian rings, locally Artinian rings, Noetherian rings, locally Noetherian rings and Nakayama's Lemma.

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