The annihilator of radical powers in the modular group ring of a 𝑝-group

We show that if N N is the radical of the group ring and L L is the exponent of N N , then the annihilator of N w {N^w} is N L − w + 1 {NL - w + 1} . As corollaries we show that the group ring has exactly one ideal of dimension one and if the group is cyclic, then the group ring has exactly one ideal of each dimension.