Nakayama lemma

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

Evidence

We accept. It is a minimal system of generators. Since it is not trivial, and follows.

By assumption, there would be with an equation of the form, ie.

There lies in the Jacobson radical, the factor is a unit. The generating system is thus not minimal and thus to rebut the presumption.

Conclusions

• If a finitely generated module, a sub-module and an ideal, it shall

This conclusion, which is equivalent to the above lemma and is therefore also known as Nakayama 's lemma, one can use for lifting of bases:

• There were a local ring, its maximal ideal and the residue field.