μ–essential – Supplemented modules

Consider M is any identity ring and let W is a unital lefting M- module. Let L be a sub.module of an – M module1W , a sub.module K in W is called a “μ* - essential supplement1” of L in W , when W= L+K and L ∩ K≪μ*K. W is called1 μ* - essential – supplemented1 module , if each sub.module in W has1 μ* - essential – supplement1 in.W. also an M- module W is called1 amplye μ* -essential – supplemented1 when any sub-module L , K in W with W = L+K , there exists μ* essential –supplement T of L contained in K. M- module W is called1 cofinitly μ* – essential – supplemented1 ( denoted by cof – essential. Supplemented in W ). If each cofinite sub-module of W has μ*– essential – supplement . and module W is called1 ⊕ μ*–essential – supplemented1 module if each sub-module in W has μ*–essential –supplement1 which is a direct- summand in W . Let T and L be sub-modules of W whenT ≤L≤W, T is called μ*e – co – essential sub-module of L in W ( denoted by T ≤μce* L in W ) , if LT≪μe*WT. A characterizations of μ* -essential – ce T e T supplemented and a conditions under which the direct sum and the finite sum of μ* −essential –supplemented , amply μ* -essential – supplemented , cofinitly μ* – essential – supplemented , and the ⊕ μ*–essential supplemented is a μ* −essential – supplemented , amply μ* -essential – supplemented , cofinitly μ* – essential – supplemented , and the ⊕ μ*–essential – supplemented respectively, are given.