1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k. 2. Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or (an)n=a.

1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k. 2. Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or (an)n=a.

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 21E: 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive...

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Question

1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k.

2. Let a be an element of a ring R such that a³=1_R. Prove: for any integer n, either (aⁿ)ⁿ=1_R or (aⁿ)ⁿ=a.

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