functional subrings - MathOverflow

functional subrings

2012-08-28T14:51:37Z

I should recall the notion of maximal subring of a commutative unitary ring $R$.

Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a commutative ring with the restricted addition and multiplication of $R$ and also $S\subsetneq T$ then we could deduce that $T=R$.

I am interested in studing this notion in Rings of continuous functions.

We could easily deduce that for $x \neq y \in X$ The set of the form $$M_{x,y}=\Big(f\in C(X): f(x)=f(y) \Big)$$ forms a maximal subring of $C(X)$

From the above summary and notations I could pose my Questions.

Question1: Is there a maximal subring in $C(X)$ other than all $M_{x,y}$'s?

Question2: Is $X$ compact if all maximal subrings of $C(X)$ is of the form $M_{x,y}$?

PS:I suppose that all subrings of a commutative ring $R$ contains the unitary element of $R$.

Answer by Josh Ree for functional subrings

2012-08-28T18:52:56Z

You're assuming that $C(X)$ actually separates the points of $X$. Otherwise one could have $M_{x,y}=C(X)$.

Answer by jbc for functional subrings

2012-08-29T10:05:46Z

Question 1 has a positive answer for general $X$. The reason for this is that $C(X)$ cannot distinguish between $X$ and its realcompactification. Hence if $X$ is not realcompact we construct $M_{x,y}$ as above with $x$ in $X$ and with $y$ in the realcompactification, but not in $X$. This is a maximal subring which does not have the required form.

A good reference for this material is Weir's book "Hewitt-Nachbin spaces" (and, of course, Gilman and Jerison).

Answer by INI for functional subrings

2012-10-19T07:48:12Z

Hi, Yes, C(X) has very different maximal subrings. First note that the real line, say R, has uncountable many maximal subrings, see for example A. Azarang and O.A.S. Karamzadeh works about the existence of maximal subrings in Fields and Commutative Rings, Hence for any fixed maximal ideal M_x of C(X), C(X) contains many maximal subrings different from the above. Also note that for any free maximal ideal M, C(X) contains a maximal subring which contains R.