Suppose is a commutative diagram of -modules and -module homomorphisms.
(a) Suppose that is injective. Prove that the map is injective.
(b) Suppose that is surjective. Prove that the map is surjective.
these maps are well-defined because of the commutativity of your diagram: if then thus also if then
thus hence: therefore so the maps and are well-defined. suppose
is inective and so from the definition of we'll get thus because is injective. if is surjective and then for some