Let $K$ be a field and let $G$ be a finite abelian group. For simplicity, we will assume that the characteristic of $K$ does not divide the order of $G$ . Let $\varphi_1,\ldots, \varphi_n$ be a complete set (up to equivalence) of distinct irreducible (linear) representations of $G$ over $K$ , so that $\varphi_i$ is a homomorphism: $$\varphi_i\colon G \longrightarrow \operatorname{GL}(n_i,K)$$ where $n_i$ is the degree of the representation $\varphi_i$ and $\sum_i n_i=|G|$ . Let $\chi_1,\ldots,\chi_n$ be the irreducible characters attached to the $\varphi_i$ , i.e. the function $\chi_i\colon G \to K$ is defined by $$\chi_i(g)=\text{Trace}(\varphi_i(g)).$$ Notice, however, that in general the map $\chi_i$ is not a homomorphism from the group into either the additive or multiplicative group of $K$ . We define a system of primitive orthogonal idempotents of the group ring $K[G]$ , one for each $\chi_i$ , by: $${\bf 1}_{\chi_i}=\frac{1}{|G|}\sum_{g\in G} \chi_i(g^{-1})g\in K[G]$$ so that $\sum_{i}{\bf 1}_{\chi_i}=1\in K$ and ${\bf 1}_{\chi_i}\cdot {\bf 1}_{\chi j}=\delta_{ij}$ where $\delta_{ij}$ is the Kronecker delta function. We define the $\chi_i$ component of $K[G]$ to be the ideal $K[G]_{\chi_i}={\bf 1}_{\chi_i}\cdot K[G]$ . Notice that $V_i=K[G]_{\chi_i}$ is a finite dimensional $K$ -vector space, on which $G$ acts. Thus, the representation of $G$ afforded by the $K[G]$ -module $V_i$ , call it $\varphi$ , must be one of the representations $\varphi_j$ defined above. Comparing the trace, one concludes that $\varphi=\varphi_i$ and $V_i=K[G]_{\chi_i}$ is a vector space of dimension $n_i$ . In particular, there is a decomposition: $$K[G]=\oplus_\chi K[G]_\chi.$$ If $k\in K[G]$ then by the previous decomposition, we can write: $$k=\sum_\chi k_\chi$$ where $k_\chi \in K[G]_\chi$ . Notice that the representations $\varphi_i$ can be retrieved as: $$\varphi_i\colon G \longrightarrow \operatorname{GL(K[G]_{\chi_i})}.$$