Something interesting I realised in my studies in Math is that certain theorems are more “useful” than others. Certain theorems’ sole purpose seem to be an intermediate step to prove another theorem and are never used again. Other theorems seem to be so useful and their usage is everywhere.
One of the most “useful” theorems in basic Ring theory is the following:
Let be a commutative ring with 1 and an ideal of . Then
(i) is prime iff is an integral domain.
(ii) is maximal iff is a field.
With this theorem, the following question is solved effortlessly:
Let be a commutative ring with 1 and let and be ideals of such that .
(i) Show that is a prime ideal of iff is a prime ideal of .
(ii) Show that is a maximal ideal of iff is a maximal ideal of .
Sketch of Proof of (i):
is a prime ideal of iff is an integral domain. ( by the Third Isomorphism Theorem. ) is a prime ideal of .
(ii) is proved similarly.