Maximal submodule

The 77-module M is called irreducible if 0 is a maximal submodule of M. Note that 0 is not an irreducible 77-module. 

From Lemma 8.3.3 we know that M contains an irreducible submodule L. Since M is assumed to be completely reducible, there exists a submodule K of M such that K L = M. According to Lemma 8.3.1, K is a maximal submodule of M, so that M has at least one maximal submodule. 

We define J(M) to be the intersection of all maximal submodules of M and call this submodule the Jacobson radical of M. 

Let us assume, by way of contradiction, that M possesses a maximal submodule L with K % L. Then K + L = M. From this we conclude that... 

A submodule L of the 17-module M different from M is called maximal if L and M are the only submodules of M containing L (as a subset). 

Note that, as a consequence of Lemma 8.3.3 and Lemma 8.3.5, each completely reducible D-module has maximal homogeneous submodules. 

Theorem 8.3.6 Assume that M is completely reducible, and let hi denote the set of all maximal homogeneous submodules of M. Then the following hold. 

Proposition 8.5.2 Assume D to be semisimple. Then each maximal homogeneous submodule of the D-module D is a minimal ideal of D. 

Let x be an irreducible character of CS. Since CS is semisimple, we obtain from Theorem 8.6.2(i) together with Theorem 8.6.4(h) that there exists exactly one maximal homogeneous submodule Hx of the OS -module CS such... 

Recall from 3.1 the definition of the maximal S-torsion submodule of an R-module M... 

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