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Abstract Algebra: An Introduction
Found in: Page 80
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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Short Answer

If R is a ring with identity, and f:RS is a homomorphism from R to a ring S , prove that f(1R) is an idempotent in S . [Idempotent were defined in Exercise 3 of Section 3.2.]

Hence it is proved that f1R is an idempotent in S .

See the step by step solution

Step by Step Solution

Property of Rings

If any ring R has elements such that, a,bR, then, addition and multiplication of the function of its elements is respectively given by:

fa+b=fa+fb fab=fafb


We have a homomorphism of rings asf:RS:

In this case, we get:

f1R=f1R1R =f1Rf1R =f1R2

Hence proved, f1R is an idempotent in S .

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