Unformatted text preview: B D B . n Exercises 8.1 In the following, R always denotes a ring and M an R –module. 8.1.1. Prove Lemma 8.1.4 . 8.1.2. Prove the assertions made in Example 8.1.16 . 8.1.3. Let I be a left ideal of R and deﬁne IM D f r 1 x 1 C ²²² C r k x k W k ³ 1;r i 2 I;x i 2 M g : Show that IM is a submodule of M . 8.1.4. Let N be a submodule of M . Deﬁne the annihilator of N in R by ann .N/ D f r 2 R W rx D for all x 2 N g : Show that ann .N/ is a (twosided) ideal of R 8.1.5. Prove Proposition 8.1.17 . 8.1.6. Prove the assertions made in Example 8.1.18 . 8.1.7. Let V be an n –dimensional vector space over a ﬁeld K , with n > 1 . Show that V is not free as an End K .V / module....
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Ring, Proof. Suppose, multiplicative identity

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