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 has no non-zeró nilpotent elements. (ii) axa - 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 ax = a. Prove the following : (i) R has no non-zeró nilpotent elements. (ii) axa - 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 .
Related questions
Question
All
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,