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(a) Prove that every ring with prime characteristic p can be embedded in a ring with unity and characteristic p. (b) Let (G,+) be an Abelian group. If n and are homomorphisms from G to G, define n+C and no by (n+C)(a) = n(a) + ((a) and no(a) = n((a)) for all a E G. Prove that n + C and no C are also endomorphisms. Also prove that with these operations the set of all endomorphisms constitutes a ring.

(a) Prove that every ring with prime characteristic p can be embedded in a ring with unity and characteristic p. (b) Let (G,+) be an Abelian group. If n and are homomorphisms from G to G, define n+C and no by (n+C)(a) = n(a) + ((a) and no(a) = n((a)) for all a E G. Prove that n + C and no C are also endomorphisms. Also prove that with these operations the set of all endomorphisms constitutes a ring.

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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Question 3. (a) Prove that every ring with prime characteristic p can be embedded in a ring with unity and characteristic p. (b) Let (G, +) be an Abelian group. If n and are homomorphisms from G to G, define n+C and no by (n+C)(a) = n(a) + ((a) and nos(a) = n(5(a)) for all a E G. Prove that n + C and no C are also endomorphisms. Also prove that with these operations the set of all endomorphisms constitutes a ring.
Transcribed Image Text:Question 3. (a) Prove that every ring with prime characteristic p can be embedded in a ring with unity and characteristic p. (b) Let (G, +) be an Abelian group. If n and are homomorphisms from G to G, define n+C and no by (n+C)(a) = n(a) + ((a) and nos(a) = n(5(a)) for all a E G. Prove that n + C and no C are also endomorphisms. Also prove that with these operations the set of all endomorphisms constitutes a ring.
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