5. Let R be the ring of all functions from the closed interval [0, 1] to real numbers (with the usual pointwise addition and multiplication). Let S be the subring of R consisting of all continuous functions. (a) Show that if f e R is not zero at any point, then f is a unit in R. (b) Show that any nonzero function g E R which is not a unit, is a zero divisor in R. (c) Give a nonzero function h e S that is neither a unit nor a zero divisor in S. (d) Give a function a E S which is a zero divisor in S.
5. Let R be the ring of all functions from the closed interval [0, 1] to real numbers (with the usual pointwise addition and multiplication). Let S be the subring of R consisting of all continuous functions. (a) Show that if f e R is not zero at any point, then f is a unit in R. (b) Show that any nonzero function g E R which is not a unit, is a zero divisor in R. (c) Give a nonzero function h e S that is neither a unit nor a zero divisor in S. (d) Give a function a E S which is a zero divisor in S.
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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![5. Let R be the ring of all functions from the closed interval [0, 1] to real numbers (with
the usual pointwise addition and multiplication). Let S be the subring of R consisting of all
continuous functions.
(a) Show that if f e R is not zero at any point, then f is a unit in R.
(b) Show that any nonzero function g E R which is not a unit, is a zero divisor in R.
(c) Give a nonzero function h E S that is neither a unit nor a zero divisor in S.
(d) Give a function a E S which is a zero divisor in S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7cb3dc1-48d8-4b46-9a7e-e05afd9f2a99%2F0faee2d8-ff53-4359-82a6-94486b4926b3%2F38fy039_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let R be the ring of all functions from the closed interval [0, 1] to real numbers (with
the usual pointwise addition and multiplication). Let S be the subring of R consisting of all
continuous functions.
(a) Show that if f e R is not zero at any point, then f is a unit in R.
(b) Show that any nonzero function g E R which is not a unit, is a zero divisor in R.
(c) Give a nonzero function h E S that is neither a unit nor a zero divisor in S.
(d) Give a function a E S which is a zero divisor in S.
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