Irreducible Modules over Associative Rings with

4 Irreducible Modules over Associative Rings with 1 

Throughout this section, the letter D stands for an associative ring with 1, the letter M for an irreducible D-module. 

Let m be an element in M 0. Since 1 mo = rn and 0 7- m, 0 7/ Dmjj. Thus, as Dtud is a submodule of M and M is assumed to be irreducible, we must have that Dmo = M. Thus, looking at the definition of Ar (m) the claim follows from Theorem 8.1.1. 

We are assuming that M is irreducible. Thus, for each element m in M 0, A i) in) is a maximal submodule of the /l-module D. Thus, by definition, J(D) C AD(M), and that means that 0 = MJ(D). 

Recall that End (M) is our notation for the ring of all D-endomorphisms of M. (We saw that End (M) is a ring with respect to componentwise addition and composition as multiplication.) 

Recall that, for each element m in M, Aoim) is a submodule of the )-module D cf. Lemma 8.1.5. 

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In Section 8.3, we focus on completely reducible modules over associative rings with 1. Section 8.4 deals with irreducible modules over these rings. In Section 8.5, we combine the results obtained in these two sections to obtain the famous (and complete) description of semisimple associative rings with 1 which was first given by Joseph Wedderburn and Emil Artin. 

D. Let / X — Y be a morphism. Assume Y is irreducible and reduced with generic point y. Let T be a quasi-coherent -module flat over oy. Then for all x G X, Tx is a torsion-free y-module. If X is noetherian and T is a coherent ox-module, this means that all associated points of T lie over y. Conversely, this property implies that T is flat over oy if all stalks oy Y are valuation rings (e.g., Y a non-singular curve, or Spec (Z)). 

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