Large images of reducible galois representations

Given a reducible Galois representation $\overline{\rho}: G_{$\mathbb{Q}$} \rightarrow GL_2( $\mathbb{F}$q)$ we show there exists an irreducible deformation $\rho : G{$\mathbb{Q}$} \rightarrow GL_2 ($\mathbb{W}$ [[T_1, T_2,…, T_r,…,]])$of$\overline{\rho}$ramifiedatinfinitelymanyprimes,where$$\mathbb{W}$$denotestheringofWittvectorsof$$\mathbb{F}$_q$. This is a modification of Ramakrishna’s result for the irreducible case.