I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says:

Let $R$ be a Noetherian local ring with maximal ideal $m$. Then there exists an $m$-primary ideal in $R$ generated by $\dim R$ elements.

In the proof that I'm reading is said that the fact follows from these $2$ lemmas:

1) Let $R$ be a Noetherian ring and $a$ an ideal in $R$ of height $r$. Assume there are elements $a_1, ..., a_{s-1} \in a$, such that the height of $(a_1, ..., a_{s-1})$ is $s-1$ for some $1 \leq s \leq r$. Then there exists an element $a_s \in a$ such that the height of $(a_1, ..., a_s)$ is $s$.

2) (Krull's Principal Ideal Theorem) Let $R$ be a Noetherian ring and $a$ an element of $R$ that is neither zero divisor nor a unit. Then every minimal prime divisor $p$ of $(a)$ is of height $1$.

So if I can find an element that is neither a unit nor a zero divisor in any Noetherian local ring, then everything is OK. But I'm not sure that every such ring has such element, i.e. is there a Noetherian local ring consisting only of units and zero divisors ? Thanks in advance !

Edit: OK, I see from the answers that there exist rings consisting only of units and zero divisors. So my question now is how to deal with them? It seems like the two lemmas are not sufficient or I'm missing something. Any ideas?


Is there a Noetherian local ring consisting only of units and zero divisors ?

Of course! Any artinian local ring consist of units and nilpotents. ($\mathbb Z/4\mathbb Z$ is such an example.)

Edit. The two lemmas are not enough to prove the theorem. Let $R=K[X,Y]_{(X,Y)}/(X^2,XY)$. Then $\dim R=1$, and the maximal ideal of $R$, that is, $\mathfrak m=(x,y)$, consists of zerodivisors since $x\mathfrak m=0$. However, there is a principal $\mathfrak m$-primary ideal, namely $\mathfrak m^2=(y^2)$.

  • $\begingroup$ Trivially, any field satisfies your requirement. $\endgroup$ – user26857 May 4 '16 at 7:45
  • $\begingroup$ Yeah, sure :D, but what about the system of parameters then ? How can the theorem be proven ? $\endgroup$ – brick May 4 '16 at 8:44
  • $\begingroup$ OK @user26857 , I'm trying to prove that every Noetherian local ring $R$ has a system of parameters (the first theorem in the question). I know how to do this if $R$ has element which is neither zero divisor nor unit, using the $2$ lemmas written in the question. However I can't deal with the case when $R$ contains only units and zero divisors, because I can't apply lemma $2)$ - Krull's Principal Ideal Theorem. So my question is how solve this problem ? $\endgroup$ – brick May 4 '16 at 9:04
  • $\begingroup$ @brick I've added an example to my answer. $\endgroup$ – user26857 May 4 '16 at 9:08
  • $\begingroup$ OK, thank you very much! Last question. Can you tell me where can I find the complete proof of the theorem, because the one I'm reading is obviously not entirely correct? $\endgroup$ – brick May 4 '16 at 9:18

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