Some Results on Modules Satisfying (C)

Let M be a unitary module over a commutative ring R with identity. We say M satisfies (C) if for every submodule of M and for every sequence (x_n) of elements of R, the ascending sequence of submodules N:x₁ ⊆ N:x_lx₂ ⊆ N:x_lx₂x₃⊆................

stabilizes: A commutative ring R with identity satisfies (C) if R satisfies (C) as an R-module.

In Section 1 of this note, we prove that every module over a commutative ring R with identity satisfies ( C) if and_ only if R is a perfect ring.

In Section 2 of this note, we determine when a module over a one -dimensional strongly Laskerian domain satisfies ( C).