In this research, we introduce a small essentially quasi−Dedekind R-module to generalize the term of an essentially quasi.−Dedekind R-module. We also give some of the basic properties and a number of examples that illustrate these properties.

Let  be a commutative ring with identity and let   be an R-module. We call an R-submodule  of  as P-essential if  for each nonzero prime submodule  of    and 0  . Also, we call an R-module  as P-uniform if every non-zero submodule  of  is P-essential. We give some properties of P-essential and introduce many properties to P-uniform R-module. Also, we give conditions under which a submodule  of a multiplication R-module  becomes P-essential. Moreover, various properties of P-essential submodules are considered.

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Let R be a commutative with non-zero identity, let A be a R-module. In this paper submodule H of an R-module A to be Soc-prime if whenever r ∈ R, a ∈ A with < a > ≤ Soc(A) and ra ∈ H then either a ∈ H or r ∈ [H: A]. We give a number of characterizations of Soc-prime submodules.