Are these statements true/ partially true/ false: a) minimal module exist in noetherian module. b) maximal submodule need not exist in an artinian module. If these are false then can you give a counter example and if partially true then please tell me about the example of the case in which it is true and in which isn't is false.
G is properly defined in example 3 in the last lecture, here: ruclips.net/video/YhfHEkXqNwg/видео.html . Now, this example with its continuation in this lecture was, with all due respect, indeed a bit confusing. Consider everything as Z-modules. Then I think what we have here is that H, defined in this lecture (Lecture 28), is a submodule of Q which indeed contains Z, so by lattice isomorphism theorem, H/Z is a submodule of Q/Z. Now H/Z is also a submodule of G < Q/Z, defined last lecture, which was shown not to be Noetherian. And there is a natural surjective homomorphism H->H/Z. Now, if H/Z is not Noetherian, then there exists some infinite chain of submodules in H/Z, which gives us an infinite chain of submodules in H, so H is not Noetherian. The problem here is the surjective homomorphism he draws at 2:50 . I'm not sure this is correct, and that he doesn't mean H -> H/Z < G, looking at the definition, and explicitly writing out what G is in the last lecture. If however, there is a surjective homomorphism H->G, then the argument holds in the same way as above, since G was shown not to be Noetherian last time. Inverse images are always (ideals) submodules, and inclusion is preserved by properties of functions and set theory alone, also surjectivity guarantees that if the pullbacked infinite chain stops, i.e. \phi^{-1}(I_k)=\phi^{-1}(I_k+1) for some k, then I_k=I_k+1 so we get a contradiction.
Are these statements true/ partially true/ false:
a) minimal module exist in noetherian module.
b) maximal submodule need not exist in an artinian module.
If these are false then can you give a counter example and if partially true then please tell me about the example of the case in which it is true and in which isn't is false.
I can't understand why H is not noetherian???
G is properly defined in example 3 in the last lecture, here: ruclips.net/video/YhfHEkXqNwg/видео.html . Now, this example with its continuation in this lecture was, with all due respect, indeed a bit confusing. Consider everything as Z-modules. Then I think what we have here is that H, defined in this lecture (Lecture 28), is a submodule of Q which indeed contains Z, so by lattice isomorphism theorem, H/Z is a submodule of Q/Z. Now H/Z is also a submodule of G < Q/Z, defined last lecture, which was shown not to be Noetherian. And there is a natural surjective homomorphism H->H/Z. Now, if H/Z is not Noetherian, then there exists some infinite chain of submodules in H/Z, which gives us an infinite chain of submodules in H, so H is not Noetherian. The problem here is the surjective homomorphism he draws at 2:50 . I'm not sure this is correct, and that he doesn't mean H -> H/Z < G, looking at the definition, and explicitly writing out what G is in the last lecture. If however, there is a surjective homomorphism H->G, then the argument holds in the same way as above, since G was shown not to be Noetherian last time. Inverse images are always (ideals) submodules, and inclusion is preserved by properties of functions and set theory alone, also surjectivity guarantees that if the pullbacked infinite chain stops, i.e. \phi^{-1}(I_k)=\phi^{-1}(I_k+1) for some k, then I_k=I_k+1 so we get a contradiction.