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Math Help - show existence of complementary submodule

  1. #1
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    show existence of complementary submodule

    I think i understand the first 2 parts of the hints but the last part of this question just confuses me =/ Prove that if N is a submodule of an R module M so that the quotientM/N is a free R module then there exists a complementary submodule to N in M. lecturer's hint: let X be a subset of M st (x + N| x in X) is basis of M/N define fn B: M/N --> M st B(x +N) = x for all x in X and show that im(B) is complementary to N
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  2. #2
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    Quote Originally Posted by gtkc View Post
    I think i understand the first 2 parts of the hints but the last part of this question just confuses me =/ Prove that if N is a submodule of an R module M so that the quotientM/N is a free R module then there exists a complementary submodule to N in M. lecturer's hint: let X be a subset of M st (x + N| x in X) is basis of M/N define fn B: M/N --> M st B(x +N) = x for all x in X and show that im(B) is complementary to N
    first note that B is well-defined because X is a basis for M/N. now let \text{Im}(B)=L and z \in N \cap L. so z=B(u), for some u \in M/N. but u=\sum c_j(x_j + N)=\sum c_j x_j + N, for some

    x_j \in X, \ c_j \in R. thus z=B(u)=\sum c_j x_j \in N. therefore u=\sum c_j x_j + N=0. so we proved that N \cap L = \{0 \}. we only now need to prove M=N+L: let y \in M. then y + N \in M/N

    and hence y+N=\sum r_jx_j + N, for some x_j \in X, \ r_j \in R. thus y - \sum r_j x_j = a \in N and so y=a+ \sum r_j x_j = a+ B(\sum r_jx_j + N) \in N + L. hence M \subseteq N + L. the other direction of

    the inclusion is trivial. this completes the proof of M=N \oplus L.
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