1. Let fi:P B, f:P2 B be epimorphisms and P1, P2 projective. Show that Pker(2) =...

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1. Let fi:P B, f:P2 B be epimorphisms and P1, P2 projective. Show that Pker(2) = P2 O ker(fi). 2. Suppose that N and N2 are submodules of an R-module M. (a) Prove that if N N, then N is an essential submodule of M if and only if N is an essential submodule of N2 and N2 is an essential submodule of M. (b) Show that N2 is an essential submodule of M if and only if N and N2 are essential submodules of M. (c) Prove that a finite intersection of essential submodules of M is an essential submodule of M. (d) Show by example that an arbitrary intersection of essential submodules of M need not be an essential submodule of M. 1. Let fi:P B, f:P2 B be epimorphisms and P1, P2 projective. Show that Pker(2) = P2 O ker(fi). 2. Suppose that N and N2 are submodules of an R-module M. (a) Prove that if N N, then N is an essential submodule of M if and only if N is an essential submodule of N2 and N2 is an essential submodule of M. (b) Show that N2 is an essential submodule of M if and only if N and N2 are essential submodules of M. (c) Prove that a finite intersection of essential submodules of M is an essential submodule of M. (d) Show by example that an arbitrary intersection of essential submodules of M need not be an essential submodule of M.