§ Nakayama's lemma
I read the statement as , when is in the jacobson radical.
- Essentially, it tells us that if a module "lives by the ", then it also "dies by the ".
- Alternatively, we factor the equation as . Since our ideal is a member of the jacobson radical, is "morally" a unit and thus . This is of course completely bogus, but cute nontheless.
- We can think of a graded ring, say acting on some graded module (say, a subideal, ). When we compute , this will bump up the grading of . If , then could not have had non-trivial elements in the first place, since the vector of, say, "non-zero elements in each grade" which used to look like will now look like . Equating the two, we get and so on, collapsing the entire ring.