Wikipedia:Reference desk/Archives/Mathematics/2016 June 5

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June 5

Noetherian ring with non-Artinian total ring of quotients

Is there a commutative Noetherian ring with a non-Artinian total ring of quotients? GeoffreyT2000 (talk) 16:06, 5 June 2016 (UTC)

How about $\mathbb {R} [x,y]/(xy=0)$? It's Noetherian by basic results: fields are Noetherian; if $R$ is Noetherian, then so is $R[x]$; quotients of Noetherian rings are Noetherian. But its ring of quotients is not Artinian, because consider $(x)\supset (x^{2})\supset (x^{3})\supset \dots$.--2406:E006:45D:1:4022:A8FC:6159:3666 (talk) 05:36, 6 June 2016 (UTC)