  We generalize the work of Ohta on the congruence modules attached to elliptic
Eisenstein series to the setting of Hilbert modular forms. Our work involves
three parts. In the first part, we construct Eisenstein series adelically and
compute their constant terms by computing local integrals. In the second part,
we prove a control theorem for one-variable ordinary $\Lambda$-adic Hilbert
modular forms following Hida's work on the space of multivariable ordinary
$\Lambda$-adic Hilbert cusp forms. In part three, we compute congruence modules
related to Hilbert Eisenstein series through an analog of Ohta's methods.
