  We study the crystalline universal deformation ring R (and its ideal of
reducibility I) of a mod p Galois representation rho_0 of dimension n whose
semisimplification is the direct sum of two absolutely irreducible mutually
non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer
groups associated with rho_1 and rho_2 we show that R/I is cyclic and often
finite. Using ideas and results of (but somewhat different assumptions from)
Bellaiche and Chenevier we prove that I is principal for essentially self-dual
representations and deduce statements about the structure of R. Using a new
commutative algebra criterion we show that given enough information on the
Hecke side one gets an R=T-theorem. We then apply the technique to modularity
problems for 2-dimensional representations over an imaginary quadratic field
and a 4-dimensional representation over the rationals.
