# Powers of ideals in group ring $\mathbb{Z}_nS_m$

Let the group ring $$\mathbb{Z}_2S_3,$$ where $$S_3$$ is the permutation group on $$3$$ elements, with presentation $$S_3 = \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau \} = \langle \sigma, \tau | \sigma^3 = \tau^2 = 1, \tau \sigma = \sigma^2 \tau \rangle$$

Let $$\mathfrak{X} = \left\{ \sum\limits_{g \in S_3} a_gg \Big| \sum\limits_{g \in S_3} a_g = 0 \right\} = \langle \sigma + \tau \rangle$$ (I guess)

My questions are:

How can I compute $$\mathfrak{X}^k$$, where $$k \ge 2?$$

There is possible generalize this method to obtain the powers $$\mathfrak{X}^k$$ of $$\mathfrak{X} = \left\{ \sum\limits_{g \in S_m} a_gg \Big| \sum\limits_{g \in S_m} a_g = 0 \right\} \subset \mathbb{Z}_nS_m?$$