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non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).
Each projective indecomposable module (and hence each projective module) in positive characteristic ''p'' may be lifted to a module in characteristic 0. Using the ring ''R'' as above, with residue field ''K'', the identity element of ''G'' may be decomposed as a sum of mutually orthogonal primitive [[idempotent]]s (
central) of ''K''[''G'']. Each projective indecomposable ''K''[''G'']-module is isomorphic to ''e''.''K''[''G''] for a primitive idempotent ''e'' that occurs in this decomposition. The idempotent ''e'' lifts to a primitive idempotent, say ''E'', of ''R''[''G''], and the left module ''E''.''R''[''G''] has reduction (mod p) isomorphic to ''e''.''K''[''G''].
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