3

Let $R$ be a ring with $1$ and $M$ a non zero left $R$-modulesuch that $M\cong M\oplus M$

Why $M$ is neither Noetherian nor Artinian?

Thanks

rschwieb
  • 153,510
siwar
  • 291
  • 1
    Do not edit your questions to be completely different after you get an answer. In fact your edit was a completely incorrect statement (it asked "is every module Noetherian?") – rschwieb Feb 06 '18 at 00:45
  • sorry i was trying to edit an other question – siwar Feb 06 '18 at 00:47
  • If $\phi : M \oplus M \to M$ is an isomorphism, then $M, \phi(M \oplus 0), \phi(\phi(M \oplus 0) \oplus 0), \ldots$ is a strictly decreasing sequence of submodules. – Daniel Schepler Feb 06 '18 at 00:57

1 Answers1

4

For Noetherian modules, every surjective endomorphism is bijective. If we compose the isomorphism $M \cong M \oplus M$ with the projection to the first coordinate, we get a non-injective surjective endomorphism. To see that it is not injective, take $m \in M$ $m \neq 0$, then the preimage of $(0,m)$ under the isomorphism $M \cong M \oplus M$ is non-zero, but is mapped to zero under the map described above.

Dually, for Artinian modules, every injective endomorphism is bijective. So if we compose the inclusion in the first coordinate $M \to M \oplus \{0\} \subset M \oplus M$ with the isomorphism $M \oplus M \cong M$, we get a non-surjective injective endomorphism. To see that it is not surjective, note that composing with an isomorphism doesn't change surjectivity, so it is not surjective because the inclusion $M \to M \oplus \{0\} \subset M \oplus M$ is not surjective.

Lukas Heger
  • 20,801
  • i didn't anderstand the two contradictions? we have a contradiction whith which hypothesis? – siwar Feb 05 '18 at 22:43
  • @siwar For every Noetherian module, a surjective endomorphism is bijective, but I've constructed a surjective endomorphism that is not injective for a module with satisfies $M \cong M \oplus M$, so such modules can't be Noetherian. (This is technically a proof by contraposition, not by contradiction). Similarly for Artinian. – Lukas Heger Feb 05 '18 at 22:45
  • @siwar if you're not familiar with these facts, see https://math.stackexchange.com/q/1202650 and https://math.stackexchange.com/q/145310 – Lukas Heger Feb 05 '18 at 22:49
  • and why the first is non-injective ? and the second is non-bijective? – siwar Feb 05 '18 at 22:49
  • can you please detail those two things more? thanks in advance – siwar Feb 05 '18 at 22:54
  • @siwar I have edited in some details. – Lukas Heger Feb 05 '18 at 22:57
  • that is perfect thanks a lot for your explanation! – siwar Feb 05 '18 at 23:02