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Math Help - Free modules

  1. #1
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    Free modules

    In each case determine if M is an free A-module. I such a case give a basis.

    a) A=Z,M=\{(0,3m+2n,m+n):m,n \in \mathbb{Z} \}, as submodule of \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}

    b) A is a conmutative ring, M=M_n(A), with the action (aM)ij=am_{ij}, for all a \in A, M \in M_n(A)
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  2. #2
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    Quote Originally Posted by roporte View Post
    In each case determine if M is an free A-module. I such a case give a basis.

    a) A=Z,M=\{(0,3m+2n,m+n):m,n \in \mathbb{Z} \}, as submodule of \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}

    b) A is a conmutative ring, M=M_n(A), with the action (aM)ij=am_{ij}, for all a \in A, M \in M_n(A)
    both are free. in part a) the basis is \{(0,3,1), (0,2,1)\} and in part b) the basis is \{e_{ij}: \ 1 \leq i,j \leq n \}, where e_{ij} \in M_n(A) is defined to have 1 in the (i,j) entry and 0 everywhere else.
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