we know that every maximal ideal in $C(X)$ is in this form:


$$M^p=\left\{\,f \in C^\*(x):\ p\in cl\_{\beta X} Z\left(f\right)\,\right\}$$


and every maximal ideal in $C^\*(X)$ is 


$$M^{\*p}=\left\{\,f\in C^\*(X):\ f^{\beta}\left(p\right)=0\,\right\}$$


and it is not necessary that 


$$ M^p \cap C^\*(X) = M^{\*p}$$


My question is:


If are prime ideals of $C(X)$ contained in $M^p$, in a one to one corresponding to that of $C^\*(X)$ contained in $M^{\*p}$ for $p\in \beta X$ ?
------------------------------------------------------------------------------------------------------------------------------------------------


$M^p$ and $M^{\*p}$ are maximal ideals respectively in $C(X)$ and $C^\*(X)$ correspond to $p$.
$\beta X$ is Ston-cech compactification of the space $X$
for terminology and notions you can refer to [here](https://books.google.com/books?hl=en&lr=&id=3j3SBwAAQBAJ&oi=fnd&pg=PA1&dq=Rings+of+continuous+functions&ots=0UmBXt62ta&sig=yhl_h8zAq9FKOJBIxZ1-1UgEyTc#). 


Edite
-----


you can find $\beta X$ at the beginning of Ch.6 and if you want more you should continue. also in Ch.7 section 7.11, it deal whit corresponding between maximal ideals specially. 



Corresponding between prime ideals in $C(X)$ and $C^*(X)$
In general, the prime ideals of $C(X)$ and $C^\*(X)$ is not in a one to one corresponding. In fact, The prime ideals of $C(X)$ contained in $M^p$ is in one to one corresponding with that of $C^\*(X)$ contained in $M^{\*p}$ if and only if $p\in \upsilon X$, where $\upsilon X$ is the Hewitt-Nachbin space (real compactification of X).
For more information one can see [here](https://books.google.com/books?hl=en&lr=&id=3j3SBwAAQBAJ&oi=fnd&pg=PA1&dq=gillman%202013%20rings&ots=0UnyQB6_y6&sig=UrMNqsHnz26MgSufm8Ug-s7F-4Y)
Chapter 8 specially p. 119, Problem 7. J and Problem 7. K.


