Let $G$ be a finite abelian group, let $L$ be any field containing the $|G|$ -th roots of unity, and let $\hat{G}$ denote the character group of $G$ with values in $L$ . For any character $\chi\in \hat{G}$ , we define $\varepsilon_\chi$ , the corresponding orthogonal idempotent of the group ring $L[G]$ , by

• $\varepsilon_\chi^2=\varepsilon_\chi$ for all $\chi$
• $\varepsilon_\chi\varepsilon_\psi=0$ for any $\chi\neq\psi$
• $\sum_{\chi\in\hat{G}}\varepsilon_\chi=1$
• $\varepsilon_\chi\cdot g=\chi(g)\varepsilon_\chi$

These orthogonal idempotents are used to decompose modules over $L[G]$ : If $M$ is such a module, then $M=\oplus_\chi (\varepsilon_\chi M)$ .