10
votes
2answers
376 views

Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...
5
votes
2answers
397 views

Isomorphic rings of functions

Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are ...
1
vote
1answer
241 views

The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set {$[a , b): a,b \in \mathbb{R} $} as it's base. If $X$ is a topological space, an upper ...
1
vote
1answer
347 views

Lifting identities of formal power series

I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
6
votes
3answers
955 views

functional subrings

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 ...
1
vote
0answers
215 views

Sums of Strongly z-ideals

In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} ...
7
votes
4answers
607 views

Quotient rings of $C(X)$

Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
5
votes
0answers
556 views

Maximal ideals of the rings of Baire- One Functions

A real function $f:X\rightarrow \mathbb{R}$ Is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...