equivalent formulation of Nakayama’s lemma
The following is equivalent to Nakayama’s lemma.
Clearly this statement implies Nakayama’s Lemma, by setting to . To see that it follows from Nakayama’s Lemma, note first that by the second isomorphism theorem for modules,
and the obvious map
is surjective; the kernel is clearly . Thus
So from we get . Since is contained in the Jacobson radical of , it is contained in the Jacobson radical of , so by Nakayama, , i.e. .
|Title||equivalent formulation of Nakayama’s lemma|
|Date of creation||2013-03-22 19:11:47|
|Last modified on||2013-03-22 19:11:47|
|Last modified by||rm50 (10146)|