On Locally Artinian Rings - Jabbar, Adil Kadir (Sulaimani Journal for Pure and Applied Sciences, 2017)
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.
Keywords: Artinian rings; locally Artinian rings; Noetherian rings; locally Noetherian rings and Nakayamas Lemma