Issues‎ > ‎vol19n1‎ > ‎

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



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.


[1] Bueso, J. L., Jara, P. and Verschoren, A. "Duality, Localization and Completion", Journal of Pure and Applied Algebra, Vol. 94, pp 127-141. (1994).

[2] Larsen, M. D. and McCarthy, P. J. "Multiplicative Theory of Ideals", Academic Press, New York and London, (1971).

[3] Wisbauer, R. "Foundations of Module and Ring Theory", Gordon and Breach Science Publishers, (1991).

[4] Burton, D. M. "A First Course in Rings and Ideals", Addison-Wesley Publishing Company, (1970).

[5] Kaplansky, I. "Commutative Rings", University of Chicago Press, (1974).

[6] Goodearl, K. R. , "Von Neumann Regular rings", Pitman Publishing Limited, (1979).

[7] Arnold, J. T. and Brewer, J. W. "Commutative Rings Which are Locally Noetherian", J. Math. Kyoto Univ. 11, pp 45-49. (1971).

[8] Heinzer, W. and Ohm, J. "Locally Noetherian Commutative Rings", Trans. Amer. Math. Soc. Vol. 158, pp. 273-284. (1971).

[9] Jabbar, A. K.: Locally Noetherian rings, Asian Transactions on Science & Technology, Vol. 5, Issue 3, pp. 1-3. (2015).

[10] Jabbar, A. K., "More Results on Almost Noetherian Domains", Iraqi Journal of sciences, pp. 53-58. supplement of (2009).

[11] Jabbar, A. K., "Almost Noetherian Domains Which are Almost Dedekind", Kurdistan Academicians Journal, Vol. 3, No. 1, part A, pp. 33-39. (2004).