In wikipedia it says:

Let $A$ be a commutative Noetherian ring with unity. Let $k$ be a field and $A$ finitely generated $k$-algebra. Then $A$ is Artinian if and only if $A$ is finitely generated as $k$-module.

Can anybody give me an example of Artinian ring which is a $k$-algebra but not finitely generated?

  • $\begingroup$ I want an example that the Artinian ring is a $k$-algebra but infinitely generated $\endgroup$ – Bamqf Dec 17 '13 at 17:29

The real numbers over the rationals $\mathbb{Q}$ is a $\mathbb{Q}$-algebra, Artinian, but certainly not finitely generated.

  • $\begingroup$ Thank you for the example, could you briefly explain why it's Artinian? $\endgroup$ – Bamqf Dec 17 '13 at 17:55
  • 1
    $\begingroup$ Artinian means it satisfies the descending chain condition on ideals. What are the ideals in the real numbers? $\endgroup$ – user2055 Dec 17 '13 at 17:57
  • $\begingroup$ Oh, I understand, those ideals are zero ideals hence satisfies the dcc $\endgroup$ – Bamqf Dec 17 '13 at 18:00

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