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Tito Omburo (talk | contribs) →Blocks and the structure of the group algebra: valuation ring, not maximal order in the sense of the article (a more general sense). |
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To obtain the blocks, the identity element of the group ''G'' is decomposed as a sum of primitive [[idempotent]]s
in ''Z''(''R''[G]), the [[center (ring theory)|center]] of the group algebra over the
''e'' is the two-sided ideal ''e'' ''R''[''G'']. For each indecomposable ''R''[''G'']-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its [[composition factor]]s also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the [[trivial representation|trivial module]] is known as the '''principal block'''.
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