Congruence modulo an ideal ($a\equiv b(\mathrm{mod}I)⟺a-b\in I$)

Two elements of a commutative ring are congruent modulo an ideal if their difference lies in that ideal, equivalently if their images in the quotient ring agree. This is an equivalence relation compatible with addition and multiplication. For $R={\mathbb{Z}}_p[[T]]$ and $I=p^{k}R$, it means every coefficient difference is divisible by $p^k$.