Let $R$ be a reduced commutative non noetherian ring of dimension $d$ and $a$ a non zero divisor. Can I say that Krull dimension of $R/(a)$ is at most $d  1$?
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Since $a$ is a nonzerodivisor of $R$, it does not lie in any of the minimal primes of $R$. Therefore any chain of primes in $R$ that all contain $(a)$ has no minimal prime in the chain and can therefore be extended to a larger chain of primes in $R$, namely, by including a prime properly contained in the smallest prime in the chain. So $\dim R/(a) < \dim R$. 

