Proof
A standard proof of the Nakayama lemma uses the following technique due to Atiyah & Macdonald (1969).
- Let M be an R-module generated by n elements, and φ : M → M an R-linear map. If there is an ideal I of R such that φ(M) ⊂ IM, then there is a monic polynomial
-
- with pk ∈ Ik, such that
- as an endomorphism of M.
This assertion is precisely a generalized version of the Cayley–Hamilton theorem, and the proof proceeds along the same lines. On the generators xi of M, one has a relation of the form
where aij ∈ I. Thus
The required result follows by multiplying by the adjugate of the matrix (φδij − aij) and invoking Cramer's rule. One finds then det(φδij − aij) = 0, so the required polynomial is
To prove Nakayama's lemma from the Cayley–Hamilton theorem, assume that IM = M and take φ to be the identity on M. Then define a polynomial p(x) as above. Then
has the required property.
Read more about this topic: Nakayama Lemma
Famous quotes containing the word proof:
“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)