Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is commutative
Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is commutative
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
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Question
Check if the solution is correct. Correct the mistakes.
If x4=x for any x in R, prove that R is commutative
Let a,b be official elements of the ring R. Then:
(a+b)4 = a4 + 4a3 b + 6a2b2 + 4ab3 +b4
Note that each term of this number is equal to either a or b (or their product), according to the condition x^4=x for any x in R.
(a+b)4 = a + b
Now consider the expression (b+a)4 :
(b+a)4 = b + a
Hence we have:
a + b = (a+b)4 = (b+a)4 = b + aThis means that for any elements a and b in the ring R, the stability of commutativity increases, that is, R is a commutative ring.
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