Question

Let R be a ring and $f: \mathbb{Z} \to R$ be a ring homomorphism. Let $x = f(1)$. (a) Prove that $x \cdot x = x$. (b) Suppose that $g: \mathbb{Z} \to R$ is another homomorphism such that $x = g(1)$. Prove that $f = g$.

Name: let r be a ring and f mathbbz to r be a ring homomorphism let x f1 a prove that x cdot x x b suppose that g mathbbz to r is another homomorphism such that x g1 prove that f g 11815
Uploaded: 2022-04-06T09:05:14-08:00
Duration: 4 min 11 s
Channel: Shyam P
Description: let r be a ring and f mathbbz to r be a ring homomorphism let x f1 a prove that x cdot x x b suppose that g mathbbz to r is another homomorphism such that x g1 prove that f g 11815

          Let R be a ring and $f: \mathbb{Z} \to R$ be a ring homomorphism. Let $x = f(1)$.
(a) Prove that $x \cdot x = x$.
(b) Suppose that $g: \mathbb{Z} \to R$ is another homomorphism such that $x = g(1)$. Prove that $f = g$.

let r be a ring and f mathbbz to r be a ring homomorphism let x f1 a prove that x cdot x x b suppose that g mathbbz to r is another homomorphism such that x g1 prove that f g 11815

Added by Gregory L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shyam P

04/06/2022

Step 1

We want to show that $x \cdot x = x$. Since $f$ is a ring homomorphism, it preserves multiplication. Thus, $f(1 \cdot 1) = f(1) \cdot f(1)$. Since $1 \cdot 1 = 1$, we have $f(1) = f(1) \cdot f(1)$. Substituting $x = f(1)$, we get $x = x \cdot x$, which is what we Show more…

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Let R be a ring and $f: \mathbb{Z} \to R$ be a ring homomorphism. Let $x = f(1)$. (a) Prove that $x \cdot x = x$. (b) Suppose that $g: \mathbb{Z} \to R$ is another homomorphism such that $x = g(1)$. Prove that $f = g$.