Systems of Parameters are exactly $R$-sequences

Question

If $(R,m)$ is a local Cohen-Macaulay ring, it is well-known that each system of parameters is an $R$-sequence.

Is any $R$-sequence (in a Cohen-Macaulay ring) a system of parameters?

I am aware of the fact that in a local ring any $R$-sequence is "part of" some system of parameter for $R$.

Thanks for answering!

Answer 1

No, it's not! A regular sequence can have less terms than $\dim R$.

However, if you ask if any maximal $R$-sequence is a sop in a local CM ring, then the answer is yes since $\operatorname{depth}R=\dim R$.

Answer 2

No. If $R$ has dimension greater than one, take any element that is not a zero-divisor. This is a regular sequence and definitely not a SOP for $R$.