Let R be a ring such that for each a e R there exists XE R such that a'x = a. Prove the following : (i) R has no non-zerò nilpotent elements. (ii) аха - a is nilpotent and so axa = a. (iii) ax and xa are idempotents.
Let R be a ring such that for each a e R there exists XE R such that a'x = a. Prove the following : (i) R has no non-zerò nilpotent elements. (ii) аха - a is nilpotent and so axa = a. (iii) ax and xa are idempotents.
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 22E: 22. Let be a ring with finite number of elements. Show that the characteristic of divides .
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![Let R be a ring such that for each a e R there exists
XE R such that ax = a. Prove the following :
(i) R häs no non-zerò nilpotent elements.
a is nilpotent and so axa = a.
(ii) aхa
(iii) ax and xa are idempotents.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffa9b535-5adb-420d-bef1-68454bca6dd6%2Ff7f0bd18-3ba9-4e69-b384-3beb0eeb48f8%2For6si0d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let R be a ring such that for each a e R there exists
XE R such that ax = a. Prove the following :
(i) R häs no non-zerò nilpotent elements.
a is nilpotent and so axa = a.
(ii) aхa
(iii) ax and xa are idempotents.
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