  The central result of this paper is a refinement of Hida's duality theorem
between ordinary Lambda-adic modular forms and the universal ordinary Hecke
algebra. Specifically, we give a necessary condition for this duality to be
integral with respect to particular submodules of the space ordinary
Lambda-adic modular forms. This refinement allows us to give a simple proof
that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is
isomorphic to the Iwasawa algebra modulo an ideal related to the
Kubota-Leopoldt p-adic L-function. The motivation behind these results stems
from Ohta's proof of the Iwasawa main conjecture over Q. Specifically, the most
general application of this argument, which employs results on congruence
modules and requires one to make some restrictive hypotheses. Using our results
we are able to extend Ohta's argument and remove these hypotheses.
