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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 4E

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