For any two homomorphisms f:X + Y and g: Y + 2 of modules over R, consider the following diagram: x x Leos Y + Z izz where i and ; stand for the identity homomorphisms. By the com- mutativity of the squares, derive two exact sequences: (a) O Ker(s) + Ker(gof) Ker(g), (b) Coker(f) + Coker (gof) + Coker(g) 0. Define a homomorphism h: Ker(g) + Coker (f) by taking h(y) = y + Im(f) Coker (f) for every y e Ker(g). Prove that this homomorphism h connects the two exact sequences (a) and (b) into a single exact sequence. For any two homomorphisms f:X + Y and g: Y + 2 of modules over R, consider the following diagram: x x Leos Y + Z izz where i and ; stand for the identity homomorphisms. By the com- mutativity of the squares, derive two exact sequences: (a) O Ker(s) + Ker(gof) Ker(g), (b) Coker(f) + Coker (gof) + Coker(g) 0. Define a homomorphism h: Ker(g) + Coker (f) by taking h(y) = y + Im(f) Coker (f) for every y e Ker(g). Prove that this homomorphism h connects the two exact sequences (a) and (b) into a single exact sequence