On the Eisenstein ideal of Drinfeld modular curves

Let $\text{goth}E(\text{goth}p)$ denote the Eisenstein ideal in the Hecke algebra $\Bbb T(\goth p)$oftheDrinfeldmodularcurve$X_0(\goth p)$ parameterizing Drinfeld modules of rank two over ${\mathbb{F}}_q[T]$ of general characteristic with Hecke level $\text{goth}p$-structure, where $\text{goth}p◃{\mathbb{F}}_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\Bbb F_q$doesnotdividetheorderofthequotient$\Bbb T(\goth p)/\goth E(\goth p)$andtheEisensteinideal$\goth E(\goth p)$ is locally principal.