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By Gregory Karpilovsky
Allow N be an ordinary subgroup of a finite team G and permit F be a box. a huge approach for developing irreducible FG-modules contains the applying (perhaps repeated) of 3 easy operations: (i) restrict to FN. (ii) extension from FN. (iii) induction from FN. this is often the `Clifford conception' constructed by means of Clifford in 1937. some time past two decades, the idea has loved a interval of full of life improvement. the rules were bolstered and reorganized from new issues of view, in particular from the point of view of graded earrings and crossed items. the aim of this monograph is to tie jointly a number of threads of the improvement for you to provide a finished photograph of the present nation of the topic. it truly is assumed that the reader has had the similar of a customary first-year graduate algebra path, i.e. familiarity with uncomplicated ring-theoretic, number-theoretic and group-theoretic strategies, and an knowing of uncomplicated homes of modules, tensor items and fields.
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It follows that el(z -- flelw) = 0 and z - flelw E Thus @z2K. m i=2 proving (2). Because flV1 C W1, it follows from (2) that But W1 is indecomposable and flV1 # 0, hence W1 = f1V1 E V1 and v = W l @ (@E2Vi) Thus ei"2v. 2! v/w1 = qz2wj and therefore, by induction, m = n and after possible recordering the Wj, V , Z W, for all i E ( 1 , . . ,m } . rn Let V be an R-module. We say that V has the unique decomposition property if the following two properties hold: (i) V is a finite direct sum of indecomposable modules.
Let V # 0 be a n R-module which i s both artinian and noetherian. T h e n V has the unique decomposition property. P ro o f. 9, V is a finite direct sum of indecomposable modules. 4). 5. w Corollary. If R is a n artinian ring, then any finitely generated R-module V # 0 has the unique decomposition property. 8. P ro o f. 15(iii). 7. 6. Group algebras Let R be a commutative ring and let G be a group, possibly infinite. Group algebras 45 The group algebra RG is the free R-module on the elements of G , with multiplication induced by that in G.
KrVr. 1, ki = ni for all i E (1,. . ,r } . ), D = Dl. and let V be an irreducible A- = module. 21, EndA(V) E Do. On the other hand, V E Vj and hence EndA(V) E EndR(V,) E D; This proves that Di S Dj and the result follows. w We close by proving the following classical result. 25. Theorem. (Wedderburn). Let A be a finite-dimensional algebra over a field F and let I be an ideal of A which has an F-basis consisting of nilpotent elements. T h e n I is a nilpotent ideal of A . Proof. We may harmlessly assume that F is algebraically closed.