Some results on modules satisfying S-strong accr∗

The rings considered in this article are commutative with identity. Modules are assumed to be unitary. Let R be a ring and let S be a multiplicatively closed subset of R . We say that a module M over R satisfies S - strong a c c r ∗ if for every submodule N of M and for every sequence r n > of elements of R , the ascending sequence of submodules ( N : M r 1 ) ⊆ ( N : M r 1 r 2 ) ⊆ ( N : M r 1 r 2 r 3 ) ⊆ ⋯ is S -stationary. That is, there exist k ∈ N and s ∈ S such that s ( N : M r 1 ⋯ r n ) ⊆ ( N : M r 1 ⋯ r k ) for all n ≥ k . We say that a ring R satisfies S - strong a c c r ∗ if R regarded as a module over R satisfies S -strong a c c r ∗ . The aim of this article is to study some basic properties of rings and modules satisfying S -strong a c c r ∗ .