Converse of a dimension lemma

Question

Consider the following lemma. It comes from the Stacks Project.

Lemma 9.59.11. Suppose that $R$ is a Noetherian local ring and $x\in\mathfrak m$ an element of its maximal ideal. Then $\dim R\le \dim R/xR+1$. If $x$ is not contained in any of the minimal primes of $R$ then equality holds. (For example if $x$ is a nonzerodivisor.)

Does the converse hold? Precisely, if equality holds above in the statement above, is $x$ not a zerodivisor? If this is not true, can we add a set of reasonable hypotheses and make it true?

(I strongly suspect it does hold, but I would like confirmation.)

Answer 1

The answer to your question is the following: the equality holds iff $x$ is part of a system of parameters. Since in a Cohen-Macaulay ring systems of parameters are regular sequences (and viceversa), then you can easily deduce your answer.

$\textbf{}$

Answer 2

According to the answer here by Steven Sam, the other direction holds if the ambient ring is Cohen-Macaulay. In particular, all regular rings are Cohen-Macaulay.

Answer 3

I think the converse does not hold. Take $R=k[[x,y]]/(x^2, xy)$, where $k$ is a field, and mod out by $y$.