Abstract—Among the important types of submodules of a module are maximal submodules, prime and primary submodules. New concepts such as, strongly and strongly irreducible submodules were introduced in last decade. Under some conditions, equality holds for these concepts. Only few conditions that satisfy the equivalency of these types of submodules are known while most of them are yet to be investigated. Some conditions were investigated following the introduction of the concepts of irreducibility and strongly irreducibility of submodules. In this present paper, first the importance of the set of zero divisors was addressed. Then, to create new condition for equivalency, finitely generated modules and multiplicative modules also are applied. Finally, it is shown that the combination of multiplicatively, being finitely generated, and faithfulness of a module is the condition for equivalency

Index Terms—Irreducible, Strongly Irreducible, Distributive and Arithmetical Modules.

N. Ahang and A. Khaksari are with Shiraz Payam-e-Noor University (tel.: +98-9171023389; e-mail: najmeh.ahang@gmail.com).
M. Zandi is with Yazd University, Islamic Republic of Iran.

Cite: Najmeh Ahang, Ahmad Khaksari, and Maryam Zandi, "Discussions on Necessary Conditions for Equivalency of Primary, Irreducible, and Strongly Irreducible Submodules," International Journal of Applied Physics and Mathematics  vol. 2, no. 5, pp. 383-387, 2012.