College Algebra Exam Review 359

College Algebra Exam Review 359 - dent over R and Œv 1;v n...

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8.4. FINITELY GENERATED MODULES OVER A PID, PART I 369 where Q A j is the matrix obtained by replacing the j –th column of A by b . 8.4. Finitely generated Modules over a PID, part I In this section, and the following section, we will determine the struc- ture of finitely generated modules over a principal ideal domain. We begin in this section by considering finitely generated free modules. Let R be a commutative ring with identity element, and let M denote an R –module. Represent elements of the R –module M n by 1 –by– n ma- trices (row “vectors”) with entries in M . For any n –by– s matrix C with entries in R , right multiplication by C gives an R –module homomorphism from M n to M s . Namely, if C D .c i;j / , then OEv 1 ; : : : ; v n Ł C D " X i c i;1 v i ; : : : ; X i c i;s v i # : If B is an s –by– t matrix over R , then the homomorphism implemented by CB is the composition of the homomorphism implemented by C and the homomorphism implemented by B , OEv 1 ; : : : ; v n Ł CB D .OEv 1 ; : : : ; v n Ł C/B; as follows by a familiar computation. If f v 1 ; : : : ; v n g is linearly indepen-
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Unformatted text preview: dent over R and Œv 1 ;:::;v n ŁC D , then C is the zero matrix. See Exer-cise 8.4.1 Let us show next that any two bases of a finitely generated free R – module have the same cardinality. Lemma 8.4.1. Let R be a commutative ring with identity element. Any two bases of a finitely generated free R –module have the same cardinality. Proof. We already know that any basis of a finitely generated R module is finite. Suppose that an R module M has a basis f v 1 ;:::;v n g and a spanning set f w 1 ;:::;w m g . We will show that m ± n . Each w j has a unique expression as an R –linear combination of the basis elements v j , w j D a 1;j v 1 C a 2;j v 2 C ²²² C a n;j v n : Let A denote the n –by– m matrix A D .a i;j / . The m relations above can be written as a single matrix equation: Œv 1 ;:::;v n ŁA D Œw 1 ;:::;w m Ł: (8.4.1)...
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