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

= 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)