End á´ª -Prime Submodules - Nuhad S. AL-Mothafar; Adwia J. Abdil -Khalik (Ibn Al-Haitham Journal for Pure and Applied Sciences, 2017)
Let R be a commutative ring with identity and M  an unitary R-module. Let ï¤(M)  be the set of all submodules of M, and ï¹: ï¤(M) ï‚¾ï‚® ï¤(M) ïƒˆ {ï¦} be a function. We say that a proper submodule P of M is end-ï¹-prime if for each ï¡ ïƒŽ EndR(M) and x ïƒŽ M, if ï¡(x) ïƒŽ P, then either x ïƒŽ P + ï¹(P) or ï¡(M) ïƒ P + ï¹(P). Some of the properties of this concept will be investigated. Some characterizations of end-ï¹-prime submodules will be given, and we show that under some assumtions prime submodules and end-ï¹-prime submodules are coincide.