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.
this is trivial! the only thing which you need to notice is the definition of the maps which are as follows: and
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
thus: