Idempotents in group algebras and minimal abelian codes

We compute the number of simple components of a semisimple finite abelian group algebra and determine all cases where this number is minimal; i.e. equal to the number of simple components of the rational group algebra of the same group. This result is used to compute idempotent generators of minimal abelian codes, extending results of Arora and Pruthi [S.K. Arora, M. Pruthi, Minimal cyclic codes of length 2pn, Finite Field Appl. 5 (1999) 177–187; M. Pruthi, S.K. Arora, Minimal codes of prime power length, Finite Field Appl. 3 (1997) 99–113]. We also show how to compute the dimension and weight of these codes in a simple way.