equivalent formulation of Nakayama’s lemma


The following is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to Nakayama’s lemma.

Let A be a ring, M be a finitely-generated A-module, N a submodule of M, and 𝔞 an ideal of A contained in its Jacobson radicalMathworldPlanetmath. Then M=𝔞M+NM=N.

Clearly this statement implies Nakayama’s Lemma, by setting N to 0. To see that it follows from Nakayama’s Lemma, note first that by the second isomorphism theorem for modules,

𝔞M+NN=𝔞M𝔞MN

and the obvious map

𝔞M𝔞MN:ama(m+N)

is surjectivePlanetmathPlanetmath; the kernel is clearly 𝔞MN. Thus

𝔞M+NN𝔞MN

So from M=𝔞M+N we get M/N=𝔞(M/N). Since 𝔞 is contained in the Jacobson radical of M, it is contained in the Jacobson radical of M/N, so by Nakayama, M/N=0, i.e. M=N.

Title equivalent formulation of Nakayama’s lemma
Canonical name EquivalentFormulationOfNakayamasLemma
Date of creation 2013-03-22 19:11:47
Last modified on 2013-03-22 19:11:47
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 13C99