Noetherian Catenary ring and Cohen-Macaulay ring
Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $p\subset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same length.
It is true that for any two prime ideals $p$ and $q$, $p\subset q$, if we have $\operatorname{ht}p+\operatorname{ht}(q/p)=\operatorname{ht}q$, then $A$ is catenary. My question is if the converse is true, that is, if $A$ is catenary then is the above condition satisfied?
Also every Cohen-Macaulay ring is catenary (in fact, universally catenary). What will be an example of a catenary ring which is not Cohen-Macaulay?
Thank you in advance.
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$\begingroup$A factor ring of a CM ring which is not CM would be an example. For instance, $R=k[x,y,z]/(xz,yz)$ for some field $k$. In the same example, let $q = (x,y,z)R$. Then height of $q$ is $2$. Take $p = (x,y)$. Then height of $p$ is zero, but height of $q/p$ is $1$.
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