Definite article before an abstract noun
Solution 1:
I'd say, I sentence 2, II sentence 2, and III sentence 1 are correct; the latter, since dropping the in this case would yield a gap in the sentence.
Solution 2:
When is the definite article the appropriate before an abstract noun?
When you are referring to a specific instance of that abstract noun, e.g. a specific idea or theorum, and aren't identifying it by e.g. following the noun with a number specific enough for the reader to know exactly which theorum you're discussing.