Lemma of Nakayama

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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 .


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.


  • 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:

Then are archetypes of a base - vector space , then generate the modulus .

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