Abstract
D is a unitary left and R is an associative ring with identity. Module R. We call a submodule C of an R-module D called purely-small (for short Pu-small), denoted by (C<<Pu D) if C + B ≠ D For every proper pure sub-module B of D. When D is finitely generated multiplication and faithful, we study A purely-small in D in this paper. Conditions when[A:D] purely-small in R are given. This paper determines to investigate some new results on the notion of purely small sub-modules introduced by Ahmad, Dhari, and Atiya various properties of this kind of sub-modules considered.
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