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

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < June 4 ! width="25%" align="center"|<< May | June | Jul >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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 $ℝ[x,y]/(xy=0)$? 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)