Faranak Farshadifar - Department of Mathematics Education, Farhangian University, P. O. Box ۱۴۶۶۵-۸۸۹, Tehran, Iran.

Let R be a commutative ring with identity and let M be an R-module‎. ‎The purpose of this paper is to introduce and investigate the submodules of an R-module M which satisfy the dual of Property \mathcal{A}‎, ‎the dual of strong Property \mathcal{A}‎, ‎and the dual of proper strong Property \mathcal{A}‎. ‎Moreover‎, ‎a submodule N of M which satisfy Property \mathcal{S_J(N)} and Property \mathcal{I^M_J(N)} will be introduced and investigated‎.

Dual of Property \mathcal{A}, Dual of proper strong Property \mathcal{A}, Dual of strong Property \mathcal{A}, Finitely generated, Property \mathcal{I^M_J(N)}, Property \mathcal{S_J(N)}