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Let R denote a commutative ring with 1 # 0. For n € Z>o, we write n E R for 1 + ... + n times An endomorphism of R is defined as a ring homomorphism from R to itself. Denote by Yn the map R→ R given by Yn(x) = x" for x € R. Prove the following. (1a) If 2 is an endomorphism, then 2 = 0 in R. (1b) If 43 is an endomorphism, then 6 = 0 in R. (1c) If 43 is an endomorphism, then it is possible that 3 / 0 in R.

Let R denote a commutative ring with 1 # 0. For n € Z>o, we write n E R for 1 + ... + n times An endomorphism of R is defined as a ring homomorphism from R to itself. Denote by Yn the map R→ R given by Yn(x) = x" for x € R. Prove the following. (1a) If 2 is an endomorphism, then 2 = 0 in R. (1b) If 43 is an endomorphism, then 6 = 0 in R. (1c) If 43 is an endomorphism, then it is possible that 3 / 0 in R.

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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Let R denote a commutative ring with 10. For n € Z≥o, we write n E R for 1 + ··· + 1. n times. An endomorphism of R is defined as a ring homomorphism from R to itself. Denote by n the map R→ R given by n(x) = x" for x = R. Prove the following. (1a) If 2 is an endomorphism, then 2 = 0 in R. (1b) If 3 is an endomorphism, then 6 = 0 in R. (1c) If 43 is an endomorphism, then it is possible that 3 / 0 in R.
Transcribed Image Text:Let R denote a commutative ring with 10. For n € Z≥o, we write n E R for 1 + ··· + 1. n times. An endomorphism of R is defined as a ring homomorphism from R to itself. Denote by n the map R→ R given by n(x) = x" for x = R. Prove the following. (1a) If 2 is an endomorphism, then 2 = 0 in R. (1b) If 3 is an endomorphism, then 6 = 0 in R. (1c) If 43 is an endomorphism, then it is possible that 3 / 0 in R.
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