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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Question

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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