Peer Review Report For: S-Pseudo Bounded Radical of Submodules [version 1; peer review: 1 not approved]

In this paper, every module M is unitary and every ring F is commutative with identity. Some properties of S-Pseudo bounded radical are studied. A proper submodule ℵ of an F − module M is said to be S-pseudo bounded submodule if there exists , φ ∈ End ( M ) , x ∈ M such that φ ( x ) ∈ ℵ implies ann F ( ℵ ) = ann F φ ( x ) making use of endomorphism map over an F − module . The characterization of S-Pseudo bounded radical for finitely generated and multiplication module is given. It must be emphasized that scalar module and prime submodule played a major role in achieving new results and to study the relationship between the radical of submodules and the radical of S-pseudo bounded submodule. In our work many properties and corollaries will be proved which explain the idea of the radical of S-pseudo bounded submodule and a new class of F − module as well as F − submodule provided with some examples that illustrate and clarify in a nice way this type of module (submodule). We used the symbol End ( M ) which means the set of all endomorphism maps of F − module M and S − rad M PS . B . ( ℵ ) refers to the intersection of all S-pseudo bounded submodules of M containing ℵ .