  We use pseudodeformation theory to study the analogue of Mazur's Eisenstein
ideal with certain squarefree levels. Given a prime number $p>3$ and a
squarefree number $N$ satisfying certain conditions, we study the Eisenstein
part of the $p$-adic Hecke algebra for $\Gamma_0(N)$, and show that it is a
local complete intersection and isomorphic to a pseudodeformation ring. We also
show that in certain cases, the Eisenstein ideal is not principal and that the
cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we
prove that "multiplicity one" fails for the modular Jacobian in these cases. In
a particular case, this proves a conjecture of Ribet.
