On essential (complement)‎ submodules with respect to an arbitrary submodule

الملخص EN

In this paper we Proved other properties of essential and complement submodules to an arbitrary submodule of an R-module M .We prove that for a family {Mα}α εʌ of modules .

If Tα and Nα are submodules of Mα with N α +T α ≤ T α-e M α ,∀ α, then ⨁αεʌ (Nα + Tα) ≤ ⨁ α εʌ Tα –e ⨁α εʌ Mα.

Also we show that for submodules T, A, B and C of a module M such that T ≤ A ≤ C.

If B is T- c for A in M and C is T- c for B in M , then C is maximal T- essential extension of A in M .