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 .
proof
We accept . Let it be a minimal generating system of . Since is nontrivial, follows and .
According to the assumption , there would then be an equation of the form with , therefore .
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.
Inferences
- If a finitely generated module, a sub-module and an ideal, then applies
- .
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 .
- Then are archetypes of a base - vector space , then generate the modulus .
Individual evidence
- ↑ 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
- ↑ Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , Lemma IV.2.2