For the exercise at At 3:54. Remember that we are considering (a)/(a^2) and A/(a) as A-modules, when trying to construct an isomorphism; See math.stackexchange.com/questions/3110460/is-a-a2-cong-r-a-or-something-else-for-ideals-a-a2-leq-r/3112034#3112034 for a proof and some useful discussion/explaination.
For the exercise at At 3:54. Remember that we are considering (a)/(a^2) and A/(a) as A-modules, when trying to construct an isomorphism; See math.stackexchange.com/questions/3110460/is-a-a2-cong-r-a-or-something-else-for-ideals-a-a2-leq-r/3112034#3112034 for a proof and some useful discussion/explaination.