Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

10

Expert-verified

Abstract Algebra: An Introduction

Found in: Page 80

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

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

Hence it is proved that $f (1_{R})$ is an idempotent in $S$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Property of Rings

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

$f (a + b) = f (a) + f (b) f (a b) = f (a) f (b)$

Homomorphism

We have a homomorphism of rings as $f : R \to S$ :

In this case, we get:

$f (1_{R}) = f (1_{R} 1_{R}) = f (1_{R}) f (1_{R}) = f {(1_{R})}^{2}$

Hence proved, $f (1_{R})$ is an idempotent in $S$ .

Most popular questions for Math Textbooks

A Boolean ring is a ring R with identity in which $x^{2} = x$ for every $x \in R$ . For examples, see Exercises 19 and 44 n section 3.1. If R is a Boolean ring, prove that

(a) $a + a = 0_{R}$ for every $a \in R$ , which means that $a = - a$ . [Hint: Expand ${(a + a)}^{2}$ .]

(b) R is commutative. [Hint: Expand ${(a + b)}^{2}$ ].

Let $a$ be a nonzero element of a ring $R$ with identity. If the equation $a x = 1_{R}$ has a solution $u$ and the equation $y a = 1_{R}$ has a solution $v$ , prove that u=v .

Show that the subset $R = {0, 3, 6, 9, 12, 15}$ of $ℤ_{18}$ is a subring. Does R have an identity?

Let $f : R \to S$ be the homomorphism of rings. If $r$ is a zero divisor in $R$ , is $f (r)$ a zero divisor in $S$ ?

Let $R a n d S$ be the ring and let $R$ be the subring of $R \times S$ consisting of all elements of the form $(a, 0_{S})$ . Show that the function $f : R \to R$ given by $f (a) = (a, 0_{S})$ is an isomorphism

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free