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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Solution Summary
The expert examines schur's lemma implying functions.
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Please see the attachment.
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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