Suppose we are given a commutative ring $R$ with a unit. Suppose that $R$ is the direct product of two rings $R\cong R_1\times R_2$. It's straightforward to show that any ideal $I\subset R$ maps to an ideal $I_1\times I_2\subset R_1\times R_2$ by the above isomorphism. It is, however, not straightforward at all to give a proper description of $I_1$ and $I_2$.

To be precise, my problem is the following, arising from the book *"The Connective K-Theory of Finite Groups"* by Bruner and Greenlees.
Let $C_n$ be the cyclic group with $n$ elements; then in odd degrees $2i-1$ we have

$ku_{2i-1}(BC_n)\cong \left(\mathbb{Z}[\alpha]/(1+\alpha+\ldots+\alpha^{n-1})\right)/(1-\alpha)^i$

I am especially interested in the case where $n=p^k$ for $p$ an odd prime and $k\geq2$. By the Chinese remainder theorem we have an isomorphism

$$\mathbb{Z}[\alpha]/(1+\alpha+\ldots+\alpha^{p^k-1})\cong\prod\limits_{\begin{array}{c}d\mid p^k\\ d>1\end{array}}\mathbb{Z}[\alpha]/\Phi_d(\alpha)$$

where $\Phi_d(\alpha)$ is the $d^{th}$ cyclotomic polynomial.

If I don't make any silly mistakes the isomorphism should be given by mapping the residue class of $\alpha$ to the tuple which has the residue class of $\alpha$ at every single entry. However, if I take $p=2$ and $k=2$ the first component of $ku_{4j+2s+1}(BC_4)$ would result to be $\mathbb{Z}/p^j$ instead of $\mathbb{Z}/p^{j+1}$.

It would be great if anyone could give me hint or a reference how to solve this issue.