# Lemma of Nakayama

The Nakayama lemma , named after the Japanese mathematician Tadashi Nakayama , is the following theorem of commutative algebra :

It is a finitely generated non-trivial - module and an ideal that in the Jacobson radical of lies. Then is .${\ displaystyle M}$ ${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {a}}}$ ${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {a}} M \ neq M}$ ## proof

We accept . Let it be a minimal generating system of . Since is nontrivial, follows and . ${\ displaystyle {\ mathfrak {a}} M = M}$ ${\ displaystyle \ {u_ {1}, \ ldots, u_ {n} \}}$ ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle n \ geq 1}$ ${\ displaystyle u_ {n} \ not = 0}$ According to the assumption , there would then be an equation of the form with , therefore . ${\ displaystyle u_ {n} \ in {\ mathfrak {a}} M}$ ${\ displaystyle u_ {n} = \ sum _ {i = 1} ^ {n} a_ {i} u_ {i}}$ ${\ displaystyle a_ {i} \ in {\ mathfrak {a}}}$ ${\ displaystyle (1-a_ {n}) u_ {n} = \ sum _ {i = 1} ^ {n-1} a_ {i} u_ {i}}$ Since it is in the Jacobson radical, the factor is a unit. The generating system is therefore not minimal and thus the assumption is refuted. ${\ displaystyle a_ {n}}$ ${\ displaystyle 1-a_ {n}}$ ## Inferences

• If a finitely generated module, a sub-module and an ideal, then applies${\ displaystyle M}$ ${\ displaystyle R}$ ${\ displaystyle N}$ ${\ displaystyle {\ mathfrak {a}} \ subset J (R)}$ ${\ displaystyle M = {\ mathfrak {a}} M + N \ \ Rightarrow \ M = N}$ .

This conclusion, which is equivalent to the above lemma and is therefore also referred to as the lemma of Nakayama, can be used to raise bases:

• Let it be a local ring , its maximal ideal and the remainder class field .${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {m}}}$ ${\ displaystyle \ kappa: = R / {\ mathfrak {m}}}$ Then are archetypes of a base - vector space , then generate the modulus .${\ displaystyle x_ {1}, \ ldots, x_ {n}}$ ${\ displaystyle \ kappa}$ ${\ displaystyle M / {\ mathfrak {m}} M}$ ${\ displaystyle x_ {i}}$ ${\ displaystyle M}$ ## Individual evidence

1. Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), sentence 2.5.24
2. Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , Lemma IV.2.2