Cycle of integral subscheme in Chow group

Question

Let $X$ be a Noetherian scheme, for each closed subscheme $Z \subset X$ let $[Z]$ denotes the algebraic cycle $$ [Z] = \sum_z \operatorname{length}(O_{Z,z}) \overline{\{z\}} $$ where $z$ is generic points of irreducible components of $Z$.

MY QUESTION: Suppose $Z$ is integral subscheme. Then is above formula equals to class $\overline{\{z\}}$ where $z$ is generic of $Z$?

My thought. We need to show $\operatorname{length}((O_{Z,z}) = 1$ as $O_{X,z}$ module. Why?

Answer 1

If $Z$ is integral, then $O_{Z,z}$ is a field, so it is a simple module over $O_{X,z}$.