Let S = C[0,1] be the set of real-valued continuous functions defined on the closed interval [0, 1], where we define f+ g and fg, as usual, by (f+g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let 0 and I be the constant functions 0 and 1, respectively. Show that (a) (S,+,) is a commutative ring with unity. S has nonzero zero divisors. (b) (c) S has no idempotents #0.1.
Let S = C[0,1] be the set of real-valued continuous functions defined on the closed interval [0, 1], where we define f+ g and fg, as usual, by (f+g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let 0 and I be the constant functions 0 and 1, respectively. Show that (a) (S,+,) is a commutative ring with unity. S has nonzero zero divisors. (b) (c) S has no idempotents #0.1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1 (c)
![1. Let S = C[0,1] be the set of real-valued continuous functions
defined on the closed interval [0,1], where we define f+ g and fg,
as usual, by (ƒ+ g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let
0 and I be the constant functions 0 and 1, respectively. Show that
(a) (S, +,) is a commutative ring with unity.
(b)
S has nonzero zero divisors.
(c) S has no idempotents #0,1.
(d)
Let a = [0,1]. Then the set T = {ƒE S\ſ(a) = 0) is a subring
such that fg, gfE T for all ƒE T and g E S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19616a01-3b9f-4ded-9626-43dcd5e68cc7%2F8e5980aa-d2e2-40a3-83fb-41651c4d202c%2F4ub5kxi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let S = C[0,1] be the set of real-valued continuous functions
defined on the closed interval [0,1], where we define f+ g and fg,
as usual, by (ƒ+ g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let
0 and I be the constant functions 0 and 1, respectively. Show that
(a) (S, +,) is a commutative ring with unity.
(b)
S has nonzero zero divisors.
(c) S has no idempotents #0,1.
(d)
Let a = [0,1]. Then the set T = {ƒE S\ſ(a) = 0) is a subring
such that fg, gfE T for all ƒE T and g E S.
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