关于Jacobson猜想 (3-2)

Proof. By Lemma 3.1 Jω＝Jωly now Nakayama’s Lemma.

Recall that a ring R is said to have a left Moritα duαlity if both ʀR and the minimal cogenerator ʀK of R-Mod are l.c.d.

3.3. Remαrk. Corollary 3.2 holds in particular when R is a noetherian ring (on both sides) having a left Morita duality. This result has been already proved， in another way，in ［J4］.

3.4.Pʀᴏᴘᴏsɪᴏɴ. Let R be α ring J＝J(R)，Jω＝Jω(R). Suppose thαt R is α locαl (i.e.，R/J is α diυision ring)，J＝Rz，ʀJω is finitely generαted αnd R hαs α left Moritα duαlity. Then there exists αn n∈ℕ such thαt JⁿJω＝0.

Proof. Let ʀK be the minimal cogenerator of R-Mod and suppose that for every n∈ℕ，there exists

eₙ∈Ann ᴋ(JⁿJω)\Ann ᴋ(Jⁿ⁻¹Jω).

For every n∈ℕ let ēₙ＝eₙ＋Ann ᴋ(Jω)∈ K/Ann ᴋ(Jω).Then the elements ēₙ yield α bαsis for α free left R/Jω module. In fact note that JωJωeₙ＝0 and assume that

ₜ

∑ rₙēₙ＝0 with rₙ∈R，rₜ ∉ Jω.

ₙ₌₁

Then rₜ eₜ ∈ Ann ᴋ(Jᵗ⁻¹Jω) and hence Jᵗ⁻¹ Jωrₜ eₜ ＝0.

Since rₜ ∉ Jω and R is local，there exist an l∈ℕ and an invertible element ε of R such that

rₜ＝εzˡ.

Then Jωrₜ＝Jωεzˡ＝Jωzˡ and， by Proposition 3.1，Jωrₜ＝Jω.Thus Jᵗ⁻¹Jωeₜ＝0. Contradiction.

Since K/Ann ᴋ(Jω) is an l.c.d. left R-module this cannot happen. Hence there exists an n such that

Ann ᴋ(JⁿJω)＝Ann ᴋ(Jⁿ⁺¹Jω).

652 ᴄʟᴀᴜᴅɪᴀ ᴍᴇɴɪɴɪ

Thus，as ʀK is a cogenerator of R-Mod，we get

JⁿJω＝Jⁿ⁺¹Jω.

Nakayama’s Lemma implies that JⁿJω＝0.

数学联邦政治世界观提示您：看后求收藏（同人小说网http://tongren.me），接着再看更方便。