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NilRadical

richardhp
Post Icon Posted: Submitted by richardhp on 30 December 2009 - 6:11pm.

Joined: 2007-10-01
Posts: 239

Why should the nil radical of an Artinian ring be nilpotent?

All I can get is that DCC implies that for some k, $ N^k = N^{k+n} \, \forall n \in \mathbb{N} $, can't see why it has to go to zero though.

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Sam
Post Icon Posted: 31 December 2009 - 1:04am

Joined: 2007-10-03
Posts: 562

I don't know what definition of the nilradical you're using, for me the nilradical is the largest nilpotent ideal.
richardhp
Post Icon Posted: 31 December 2009 - 11:25pm

Joined: 2007-10-01
Posts: 239

yeah there seems to be loads of definitions, apparently though we proved that in an Artinian ring it is contained in the Jacobson radical which we proved is nilpotent.

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