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Nakayama's lemmaRecall that an ideal m in a ring A is called a maximal ideal if m = A and is maximal among the ideals in A different from A, or what amounts to the same, if A/m is a field.Theorem 1.1 (Krull). Any ring A which is different from zero contains a maximal ideal.Proof. The nonempty set of ideals in A different from A is easily seen to be inductively ordered (by inclusion). Conclusion by Zorn's lemma.Definition 1.2. A local ring is a ring A which contains precisely one maximal ideal. The residue field of the local ring A with respect to its maximal ideal is called the residue field of A.Let A be a local ring with maximal ideal m. It follows immediately from 1.1 that the elements of A outside m are invertible. Conversely, suppose given a ring A and a maximal ideal m such that all elements of A outside m are invertible, then A is a local ring. Lemma 1.3 (Nakayama's lemma). Let A be a local ring with maximal ideal m and M a finitely generated A-module. Then M/mM = 0 implies M = 0. Proof. We shall give two proofs of this lemma. Suppose M = 0 is a finitely generated A-module. Note that the set of proper submodules of M is inductively ordered and whence by Zorn's lemma contains a maximal element, say, N . The module M/N being a simple module is annihilated by m, whence mM ⊆ N = M . 1 Lecture Notes on Local Rings Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 04/13/15. For personal use only.

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