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    Schur's Lemma Implies Functions

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    I have some trouble understanding the solution to the attached problem (solution included). Could you please provide some clarification of the solution. I have indicated what my points of concern are.

    Show that if M1 and M2 are irreducible R modules, then any nonzero R-module homomorphism from
    M1 to M2 is an isomorphism. Deduce that if M is irreducible then EndR(M) is a division ring (this result
    is called Schur's Lemma). [Consider the kernel and the image.]
    Let j : M1 ! M2 be a nonzero R-module homomorphism. Then ker j 6= M and Imj 6= 0. But ker j and
    Imj are submodules of M1 and M2 respectively, so irreducibility implies ker j = 0 and Imj = M2, and
    thus j is an isomorphism.
    If M is irreducible then the above implies that any j 6= 0 in the ring EndR(M,M) has a multiplicative
    inverse, so EndR(M,M) is a division ring. 
    Please clarify the following:
    1. Why does it follow from 'Let j : M1 ! M2 be a nonzero R-module homomorphism.' that then
    ker j 6= M and Imj 6= 0.
    2. Why does irreducibility imply ker j = 0 and Imj = M2, and thus j is an isomorphism.
    3. If M is irreducible then why does the above imply that j 6= 0 in the ring EndR(M,M) has a multiplicative
    inverse, so EndR(M,M) is a division ring.

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    https://brainmass.com/math/linear-transformation/schurs-lemma-implies-functions-229856

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    Clarifications:
    1. Since is a non-zero -homomorphism, then and . The reason is as follows.
    If , then and thus is a zero -homomorphism. This is a contradiction to the condition ...

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    The expert examines schur's lemma implying functions.

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