Question

C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?

Answer 1

C(0, 1) R

(a) R is ring:

(1)

Sum of two continuous functions on [0,1] is again a continuous function on [0,1]

(2) Scalar multiplication of continuous functions on [0,1] is again a continuous function on [0,1].

(3) There is identity map that is continuous on [0,1]

(4) Inverse of continuous function continuous functions on [0,1] is a continuous function on [0,1]

Since asociative and distributive property holds in R, holds in C[0,1].

Hence R is ring

$I_c=\{f\in R:f(c)=0\}$ is an ideal :

Let

(c)g(c)0 (f +g)(c)f(c)g(c)0

Let

$f\in I_c,k\in R\\k f(c)=k*0=0\Rightarrow kf\in I_c$

Ic is an ideal.

(b) $I_{c_1}\cup I_{c_2}$ is not an ideal since

Let $f\in I_{c_1}\cup I_{c_2}\mbox{ but}f\not\in I_{c_2}$

      $g\in I_{c_1}\cup I_{c_2}\mbox{ but}g\not\in I_{c_1}$

Then

(f + g) (c) f(q) + g(c) = 0 + g(c)メ0

$I_{c_1}\cap I_{c_2}$ is an ideal

Let

$f,g\in I_{c_1}\cap I_{c_2}$

C1

Let

$f \in I_{c_1}\cap I_{c_2},k\in R\\ kf(c_1)=0,kf(c_2)=0\\ \Rightarrow kf\in I_{c_1}\cap I_{c_2}$

Hence $I_{c_1}\cap I_{c_2}$ is an ideal .