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College Algebra Exam Review 342

College Algebra Exam Review 342 - B D B n Exercises 8.1 In...

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352 8. MODULES Lemma 8.1.29. Let R be any ring with multiplicative identity, and let M be a free R –module. Any basis of M is a minimal generatiing set. Proof. Suppose B is a basis of M and that B 0 is a proper subset of B Let b 2 B n B 0 . If b were contained in RB 0 , then b could be expressed as a R –linear combination of elements of B 0 , contradicting the linear indepen- dence of B . Therefore b 62 RB 0 , and B 0 does not generate M . n Lemma 8.1.30. Let R be any ring with multiplicative identity. Any basis of a finitely generated free R –module is finite. Proof. Suppose that M is an R –module with a (possibly infinite) basis B and a finite generating set S . Each element of S is a linear combination of finitely many elements of B . Since S is finite, it is contained in the span of a finite subset B 0 of B . But then M D span .S/ span . span .B 0 // D span .B 0 / . So B 0 spans M . By the previous lemma,
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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 (two-sided) 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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