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(1d) If 6 is an endomorphism and R is an integral domain, then R is unique up to isomorphism. (1e) There are at least two non-isomorphic integral domains R for which 15 is an endomorphism. (1f) There are at least two non-isomorphic commutative rings R for which 6 is an endomorphism. Attention: in (1e) and (1f) you should not just give two such R, you also need to prove that they are not isomorphic.

(1d) If 6 is an endomorphism and R is an integral domain, then R is unique up to isomorphism. (1e) There are at least two non-isomorphic integral domains R for which 15 is an endomorphism. (1f) There are at least two non-isomorphic commutative rings R for which 6 is an endomorphism. Attention: in (1e) and (1f) you should not just give two such R, you also need to prove that they are not isomorphic.

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 1 ‡ 0. 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 Yn(x) = xª for x € R. Prove the following. (1d) If 6 is an endomorphism and R is an integral domain, then R is unique up to isomorphism. (le) There are at least two non-isomorphic integral domains R for which 415 is an endomorphism. (1f) There are at least two non-isomorphic commutative rings R for which 46 is an endomorphism. Attention: in (1e) and (1f) you should not just give two such R, you also need to prove that they are not isomorphic.
Transcribed Image Text:Let R denote a commutative ring with 1 ‡ 0. 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 Yn(x) = xª for x € R. Prove the following. (1d) If 6 is an endomorphism and R is an integral domain, then R is unique up to isomorphism. (le) There are at least two non-isomorphic integral domains R for which 415 is an endomorphism. (1f) There are at least two non-isomorphic commutative rings R for which 46 is an endomorphism. Attention: in (1e) and (1f) you should not just give two such R, you also need to prove that they are not isomorphic.
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