  Ribet has proven remarkable results about non-optimal levels of residually
reducible Galois representations. We focus on a non-optimal level $N$ that is
the product of two distinct primes and where the Galois deformation ring is not
expected to be Gorenstein. We prove a Galois-theoretic criterion for the
deformation ring to be as small as possible -- that is, for there to be a
unique newform of level $N$ with reducible residual representation. When this
criterion is satisfied, we deduce an $R=\mathbb{T}$ theorem.
