(5) Recall that an element of a ring is idempotent if a² = a. (a) Show that any integral domain can have only 1 and 0 as idempotent elements. (b) Is the converse to the previous statement true? Hint: Consider the ring of continuous functions defined on the interval [0, 1], which we call C([0, 1]). What are the idempotent elements? Is this ring an integral domain?
(5) Recall that an element of a ring is idempotent if a² = a. (a) Show that any integral domain can have only 1 and 0 as idempotent elements. (b) Is the converse to the previous statement true? Hint: Consider the ring of continuous functions defined on the interval [0, 1], which we call C([0, 1]). What are the idempotent elements? Is this ring an integral domain?
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) Recall that an element of a ring is idempotent if a² = a.
(a) Show that any integral domain can have only 1 and 0 as idempotent
elements.
(b) Is the converse to the previous statement true?
Hint: Consider the ring of continuous functions defined on the interval
[0, 1], which we call C([0, 1]). What are the idempotent elements? Is
this ring an integral domain?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8084a024-ebf9-4fb7-8600-c7fa04d2572b%2Fee163058-6a04-4f70-9332-d7f4bcfc34fe%2Fesqk1v58_processed.png&w=3840&q=75)
Transcribed Image Text:(5) Recall that an element of a ring is idempotent if a² = a.
(a) Show that any integral domain can have only 1 and 0 as idempotent
elements.
(b) Is the converse to the previous statement true?
Hint: Consider the ring of continuous functions defined on the interval
[0, 1], which we call C([0, 1]). What are the idempotent elements? Is
this ring an integral domain?
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