Appendix C: Zorn's Lemma

1. Let KM be modules, M finitely generated, say

img

If img we must show that img contains maximal members. Suppose {Xi img iI} is a chain from img and put U = ∪ iIXi . It is clear that U is a submodule and that KU, and we claim that UM . For if U = M then each xiU, and so each xiXk for some k . Since the Xi form a chain, this means that {x1, x2, . . ., xn} ⊆ Xm for some m . Since the xi generate M this means that MXm, contradicting the fact that img This shows that img and so U is an upper bound for the Xi . Hence img has maximal members by Zorn's lemma, as required.
2. Let KM be modules.
a. Let is a submodule and KX = 0} . Then is nonempty because so let {

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