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What is the name given to (if there exists any) commutative rings $R$ with identity such that $R/(a)$ is finite for every non-zero $a\in R$

Thanks a lot

drhab
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Amr
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  • They certainly exist (see $\mathbb{Z}$, or any finite ring), but I don't know of any name. – mdp Feb 26 '14 at 14:36
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    Equivalently, $R/I$ is finite, for all ideals $I\ne 0$. – Karl Kroningfeld Feb 26 '14 at 14:37
  • @MattPressland Sure. $F[x]$ where $F$ is a field is another example. I believe any euclidean domain as well. – Amr Feb 26 '14 at 14:37
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    Residually finite. – Bill Dubuque Feb 26 '14 at 14:38
  • I don't think they've received special attention, unlike residually finite groups. The larger class of rings of Krull dimension 1 have been widely studied. – arsmath Feb 26 '14 at 14:43
  • According to this question, such rings must be integral domains: http://math.stackexchange.com/questions/126206/prove-that-an-infinite-ring-with-finite-quotient-rings-is-an-integral-domain – Must Feb 26 '14 at 14:57
  • Pete L. Clark discusses them briefly in the first few pages of his notes on factorization - see his discussion of finite norm rings. This is a bit of folklore that has not yet made it into common textbooks (though I vaguely recall it appearing as an exercise somewhere). – Bill Dubuque Feb 26 '14 at 15:13

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