# Ambiguity in the definition of unmixed ideal

Compare the definitions:

Page 136 Matsumura, Commutative ring theory:

A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal.

This means that $\forall P\in {\rm ass}(I) ,\quad ht I = ht P\quad (1)$

Page 59 Bruns Herzog, Cohen-Macaulay rings, 1998:

One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors, or in modern language if the associated prime ideals of $R/I$ are the minimal prime ideals of $I$.

This means that ${\rm Min}({\rm ass}(I)))={\rm ass}(I)\quad (2)$

Clearly $(1)$ implies $(2)$ but the reverse isn't true. How these definitions are equivalent? I mean while (2) doesn`t imply (1) then how can we say the two definitions are the same?

• maybe because: both Matsumura & Bruns Herzog use unmixedness to characterize Cohen-macaulay rings. and in Cohen-macaulay rings two definitions are equal Dec 5 '14 at 16:16
• As @11156 mentioned, this is not true without additional assumption. Take $k[x,y,z]$ and $I = (x) \cap (y,z)$. So, what you stated is correct. Dec 5 '14 at 18:58