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
icon
Related questions
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.

Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning