The dual notion of the prime radical of a module

In this article, we study the second radical of a module over an arbitrary ring R as the dual notion of the prime radical of a module. We give some properties of the second radical and determine the second radical of some modules. We define the notion of m⁎-system and describe the second radical of submodules in terms of m⁎-systems. We investigate when the second radical of a module M is equal to the socle of M. In particular, we give a characterization of the socle of a noetherian module over a ring R such that the ring R/P is right artinian for every right primitive ideal P by using the concept of second radical. We also give a characterization of right quasi-duo artinian rings by using the second radical of an injective module.