1
vote
1answer
39 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
8
votes
2answers
75 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
3
votes
1answer
55 views

Painting $\mathbb R^+$ with two colors which sum of two same color numbers be the same.

Can any one paint $\mathbb R^+$ with two colors which sum of two numbers with the same color has the same color. Additional condition: Both colors should be used. I tried use Cauchy functions like ...
1
vote
1answer
131 views

Finite generation of ideal in function ring

Let $R$ be the ring of continuous functions from $[0,1]$ to the real numbers. Fix $c \in [0,1]$ and let $M_c$ = ker $E_c$ where $E_c$ denotes evaluation at $c$, a ring homomorphism from $R$ to the ...
42
votes
1answer
1k views

What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...