A note on the Grothendieck ring of the symmetric group

Let $p$ be a prime number and let $n$ be a non-zero natural number. We compute the descending Loewy series of the algebra $R\mathrm{\_}n\mathrm{/}pR\mathrm{\_}n$, where $R\mathrm{\_}n$ denotes the ring of virtual ordinary characters of the symmetric group $S\mathrm{\_}n$.