By a projector of a module X over R ,we mean an endomorphism p of the module X satisfying   pop=p . Let X be a module over R which is direct sum of family Ƒ={Xi/i∈ M} Of submodules of X. Prove that there exist a unique family {pi:X→X/i∈ M } Of projector of X such that pi(X)=Xi and pi(Xj)=0 whenever i ≠ j, Also show for every element x∈X , pi(X)=0 holds for all except atmost a finite number of indices i∈ M and we have    x= Σi∈ M pi(X

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