Basis for Quotient Ring of Group Ring

Question

I recently read a paper from passman "Observation On Group Rings" and I came across a sentence below:

Let $H\lhd G$ and let $ B=\{B_1,B_2,B_3,...\}\subseteq K[H]$ be a $K-$basis for $w(K[H])^n/w(K[H])^{n+1}$.

Here, $K[H]$ denotes a group ring of group $H$ over field $K$, $w(K[H])$ denotes the augmentation ideal of $K[H]$ and $w(K[H])^n$ denotes the $n^{th}$ power of $w(K[H])$.

What I don't understand is why the basis of the quotient ring $w(K[H])^n/w(K[H])^{n+1}$ is a subset of $K[H]$.

Answer 1

It's not a quotient ring; it's a quotient of ideals (so it forms a representation). Select coset representatives for each element of a basis of the quotient.