Please help! Thank you so much!!! 1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove that every free module is projective. Hint: Let X be a basis for F and find a map ф : X-, A. Use the universal property to get the map h you want. Then use the universal property again to show that gh = (c) Let R be a ring and let P and P2 be R-modules. Prove that P @f3 is projective if and only if Pi and P2 are projective. 1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove that every free module is projective. Hint: Let X be a basis for F and find a map ф : X-, A. Use the universal property to get the map h you want. Then use the universal property again to show that gh = (c) Let R be a ring and let P and P2 be R-modules. Prove that P @f3 is projective if and only if Pi and P2 are projective.