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