The theory of algebraic codes is a perfect illustration of the fact that the more mathematical structure one is able to add to a system, the better are the descriptions one obtairm: Changing from codes to linear codes, it is no longer necessary to compute all distances between any two codewords, only the ¢odeword weights, in order to find the minimum distance, Going from linear codes to cyclic codes then, the linear structure is replaced by a much richer algebraic structure, arm general results on properties of codes such as the Bell-bound, fast decoding algorithms, etc., may be developed.Thus it was an important result reldizing that shortened Generalized Rtmd-Muller (GR.M) codes at, cyclic codes, since the GRM codes themselves were constructed only as linear codes. And it was interesting when S.D. Berman (see [1]) in 1967 discovered that the Reed-Muller codes over GF(2) may be described v.s ideals in a very natural algebra, namely the group algebra over an elementary abclian 2-group. Based on some properties of the GRM codes discovered by T. Kasami et al. in 1968 (see [7]), E Charpm then could prove (see [2]) that a similar fact holds over GF(p).In retrospect, one is left with the impression that it is an extensive and complicated piece of work to establish these facts. Here, we go the opposite way and start with the algebra, develop some general properties, focus on some very basic ideals which happen to be the GRM codes. Our approach is very natural and is based on some classical results on group algebras developed by S. Iennings (see [6]) in the early forties, which P. Charpin apparently was unaware of. These results are straightforward when the underlying group is elementary ahelian, and may he developed from scratch.The same methods apply to not only the extended Reed-Solomon codes but to all GRM codes over arbitrary finite fields.While the first sections may be of most interest to representation theory, our last section contains a new decoding algorithm of the Reed-Muller codes, which takes advantage of the underlying algebra structure, thus justifying the whole approach. There are definitely methods there, which may be applied to other group codes, such as those based on algebraic geometry.