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# ahw1 - Math 5126 Wednesday January 23 First Homework...

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Math 5126 Wednesday, January 23 First Homework Solutions 1. Since R / I is a Noetherian ring, it is Noetherian as an R / I -module. The submodules of R / I considered as an R -module are precisely the same as the submodules of R / I considered as an R / I -module; thus R / I is a Noetherian R -module. Obviously I is a Noetherian R -module, because | I | is finite. Therefore R is a Noetherian R -module ( N and M / N are Noetherian if and only if M is Noetherian) and the result is proven. 2. If M is free (and nonzero), then M has a submodule isomorphic to R and so certainly M is not a torsion module. Conversely suppose M is not free. Since M is a nonzero cyclic R -module, it is isomorphic to R / I for some ideal I of R (reason: if M = Rm , then we can define an R -module epimorphism R M by r rm , and then we let I be the kernel). Note that I = 0 because M is not free, so there exists 0 = x I . Then x ( R / I ) = 0 which shows that M is a torsion module.
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