I just started learning college mathematics and one of the things I don't like is giving proofs by counterexamples. My question is how is disproving by giving counterexample is seen by advanced 'mathematicians', is it a bit of an 'easy way out' when we are looking to prove and disprove something. Shouldn't we look for a general structure because of which the statement is not true?
For eg. while proving that every subring of a Notherian ring may not be Notherian, I never saw a general proof, all I saw a bunch of counterexamples. Shouldn't we look for what reason a general subring of Notherian ring fails to be Notherian?