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EXAMPLE. Let A be an abelian group and let End A be the set of endomorphisms $f: A \to A$. Define addition in End A by $(f+g)(a) = f(a) + g(a)$. Verify that $f+g \in$ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity $1_A: A \to A$.

Name: EXAMPLE. Let A be an abelian group and let End A be the set of endomorphisms f: A → A. Define addition in End A by (f+g)(a) = f(a) + g(a). Verify that f+g ∈ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity 1A: A → A.
Uploaded: 2022-05-06T05:08:30-08:00
Duration: 4 min 43 s
Channel: Shaiju T
Description: EXAMPLE. Let A be an abelian group and let End A be the set of endomorphisms f: A → A. Define addition in End A by (f+g)(a) = f(a) + g(a). Verify that f+g ∈ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity 1A: A → A.

          EXAMPLE. Let A be an abelian group and let End A be the set of endomorphisms $f: A \to A$. Define addition in End A by $(f+g)(a) = f(a) + g(a)$. Verify that $f+g \in$ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity $1_A: A \to A$.

EXAMPLE. Let A be an abelian group and let End A be the set of endomorphisms f: A → A. Define addition in End A by (f+g)(a) = f(a) + g(a). Verify that f+g ∈ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity 1A: A → A.

Added by Andrea B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

05/06/2022

Step 1

Step 1: Define addition in End A as (f+g)(a) = f(a) + g(a). Show more…

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Example: Let A be an abelian group and let End A be the set of endomorphisms f:A->A. Define addition in End A by (f+g)(a) = f(a) + g(a). Verify that f+g ∈ End A. Since A is abelian, this makes End A an abelian group. Let multiplication in End A be given by composition of functions. Then End A is a (possibly noncommutative) ring with identity 1: A->A.