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How can I prove this:

Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak m^2$, $x\ne 0$, the result does not hold anymore.

I don't know how to begin! Thanks to everyone who helps!

But if $d=\infty$?

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    If $d=\infty$ then the ring is no longer Noetherian as $m$ isn't finitely generated, so all bets are off. But I'm guessing your definition of "local" requires that the ring be Noetherian.2012-01-06
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    Dear @Alex, The definition of a regular ring includes the condition of Noetherianness.2014-08-19

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Hint: Let $d$ be the Krull dimension of $A$. Show that $x$ is part of a set $x,x_2,\ldots,x_d\in A$ such that $(x,x_2,\ldots,x_d) = \mathfrak m$ iff $x\in \mathfrak m-\mathfrak m^2$. From this (and using the fact that there is a bijection between the maximal ideals of $A$ containing $(x)$ and the maximal ideals of $A/(x)$), you can see that $A/(x)$ is a local ring with maximal ideal $(x_2,\ldots,x_d)$, which has Krull dimension at most $d-1$ by Krull's Principal Ideal Theorem. The dimension is in fact exactly $d-1$, as if we had fewer than $d-1$ generators for $\mathfrak m/(x)$ then adjoining $x$ gives you fewer than $d$ generators for $\mathfrak m$, violating Krull's PIT. Since $\mathfrak m/(x)$ is generated by $d-1$ elements, this makes $A/(x)$.

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    Using this lemma can i directly say that the ring $A=K[x,y,z]_{(x,y,z)}/(x-y^2-z^2)$ is local regular and $B=K[x,y,z]_{(x,y,z)}/(x^2-y^2-z^2)$ is not, right?2012-01-05
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    One more doubt. So we can prove that if $x \in m^2$ then $A/(x)$ is not regular, or sometimes we can have $x\in m^2$ and still $A/(x)$ regular?2012-01-05
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    @balestrav As per your first comment, you can show that $A$ is local regular using this by showing $x-y^2-z^2\in m-m^2$, but not that $B$ is not. This gets into your second comment. I do not know if there is any counterexample to $x\in m^2\implies A/(x)$ not regular local, but I don't see a proof either. All that the question seems to ask is to prove that $A/(x)$ need not be regular local. This doesn't go against my hint, because the first part of the hint is both a sufficient and necessary condition for existence of a system of parameters, but what follows are just sufficient conditions for2012-01-05
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    the ring $A/(x)$ to be regular local. However, it should suggest to you some examples to show that $A/(x)$ need not be regular local if $x\in m^2$. Technically, you only need to show "if" for the first part, not "only if".2012-01-05
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    That's what i thought! So how can i prove B not to be regular. I have trouble finding the dimension of that ring, in fact i would need a primary decomposition of the ideal $(x^2-y^-z^2)$. Any ideas?2012-01-06
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    Do you need to use $B$ specifically as your example? How about $K[x]_{(x)}/(x^2)$?2012-01-06
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    The fact is that an exercize asks me specifically about that ring..2012-01-06
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    I think i've proven the Krull dimension to be less than 2, while the dimension of $m/m^2$ is 3, so it's not regular..2012-01-06
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    I think when you prove $A/(x)$ has dim d-1, your actually assume $A/(x)$ is regular local ring, otherwise you can not make sure dim$A/(x)$ is equal to the number of minimal generators.2016-02-20