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19. Assume (R,+,) is a ring with the property that a²+ a€ cent R for every element a in R. Show that (R,+,) is a commutative ring. [Hint: Make use of the expression (a + 6)2 + (a + b) to prove, first, that a. b+6.a lies in the center.)

19. Assume (R,+,) is a ring with the property that a²+ a€ cent R for every element a in R. Show that (R,+,) is a commutative ring. [Hint: Make use of the expression (a + 6)2 + (a + b) to prove, first, that a. b+6.a lies in the center.)

Advanced Engineering Mathematics
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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19. Assume (R, +,) is a ring with the property that a² + a E cent R for every element a in R. Show that (R, +, ) is a commutative ring. [Hint: Make use of the expression (a + b)2 + (a + b) to prove, first, that a . b+b.a lies in the center.) 20. Illustrate Theorem 3-18 by considering the rings (Zo, +o, e), (Z3, +3, a), and the homomorphism f: Ze - Za defined by f(0) = f(3) = 0, (1) = 1(4) = 1, S(2) - S(5) = 2. %3D %3D %3D
Transcribed Image Text:19. Assume (R, +,) is a ring with the property that a² + a E cent R for every element a in R. Show that (R, +, ) is a commutative ring. [Hint: Make use of the expression (a + b)2 + (a + b) to prove, first, that a . b+b.a lies in the center.) 20. Illustrate Theorem 3-18 by considering the rings (Zo, +o, e), (Z3, +3, a), and the homomorphism f: Ze - Za defined by f(0) = f(3) = 0, (1) = 1(4) = 1, S(2) - S(5) = 2. %3D %3D %3D
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