## Maximal *ideal and *maximal ideal

I begin to study the *operator on the domain, maybe I do not choose a perfect book which contains many theorems but no enough examples, if you know some good reference, recomment it to me please.

Now let us match the *operator with the ideal, we can get two concepts: the maximal *ideal(the maximal element of the *ideal) and * maximal ideal(*ideal and also maxiaml ideal). Here are my questions:

1) Obviously, a *maximal ideal must be a maximal *ideal, but does a maximal *ideal must be a *maximal ideal? If not, does it must be prime?

2) For any domain, does maximal *ideal or *maximal ideal must be exist?

-
 I can't tell what the question is asking, but I don't think it's about C*-algebras. Anyway, for C*-algebras every closed ideal is automatically stable under the involution. – Nik Weaver Nov 30 at 14:19 My first guess was that the stars in the question are to be understood in the sense of non-standard analysis. Unfortunately, that didn't make the question understandable. I see two ways the OP might get an answer for this question: (1) Wait until someone comes along who can correctly guess what the question means. (2) Tell us what the question means. – Andreas Blass Nov 30 at 15:15 Oh, it is not about C* algebra, the * operator is defined on fractional ideal for a domain, which is a axiom definition, double inverse is an example. I do not find it at wiki, in fact, this theory is developed after 1990. I read this topic in a Chinese book which is not a prefect book, I want to know more refenences. – Strongart Dec 3 at 11:14 Even with Strongart's comment, I still don't know what the question means. Voting to close. – Andreas Blass Dec 6 at 18:23