Nakayama lemma

Nakayama's Lemma is an important technical lemma in commutative algebra and algebraic geometry. It is a consequence of the Cayley-Hamilton theorem. One of its many equivalent statements is as follows:

Let R be a commutative ring with identity 1, I an ideal in R, and M a finitely-generated module over R. If IM=M, then there exists <math>r\in R<math>, <math>r\equiv 1 (\hbox{mod} I)<math> (?), such that rM=0. Furthermore, if I is contained in the Jacobson radical of R, then necessarily M=0.

References

• Atiyah, M.F. and Macdonald, I.G (1969). Introduction to Commutative Algebra. Addison-Wesley, Reading, MA.