Fix a prime $p > 2$. Let $rho : text{Gal}(overline{mathbb{Q}}/mathbb{Q}) to text{GL}_2(mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $mathbb{I}$ of Hidas $p$-ordinary Hecke algebra. Assume the residual representa
tion $bar{rho}$ is absolutely irreducible. Under a minor technical condition we identify a subring $mathbb{I}_0$ of $mathbb{I}$ containing $mathbb{Z}_p[[T]]$ such that the image of $rho$ is large with respect to $mathbb{I}_0$. That is, $text{Im} rho$ contains $text{ker}(text{SL}_2(mathbb{I}_0) to text{SL}_2(mathbb{I}_0/mathfrak{a}))$ for some non-zero $mathbb{I}_0$-ideal $mathfrak{a}$. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to $mathbb{Z}_p[[T]]$. Our result is an $mathbb{I}$-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.
