On Jointly Second (<i>R</i>, <i>S</i>)-Submodules

Let R and S be commutative rings and M be an (R, S)-module. In this paper, we present the dual notion of jointly prime (R, S)-submodules, that is called jointly second (R, S)-submodules, and we investigate some properties of them. We give a necessary and sufficient condition for an (R, S)-submodule being jointly second (R, S)-submodules. Moreover, we present the definition of jointly second (R, S)-modules and present a condition for jointly prime (R, S)-modules being jointly second (R, S) modules and vice versa.