/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Let \(R\) be a ring with unity \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 6

Let \(R\) be a ring with unity \(u\). Prove that the units of \(R\) form a group under the multiplication of the ring.

Short Answer

Expert verified
The set of units in a ring \(R\) with unity \(u\) does form a group under the multiplication of the ring, since it satisfies the group axioms. These units are closed under multiplication, the multiplication operation is associative, the unity of the ring serves as the group identity, and every unit has a multiplicative inverse in the ring.

Step by step solution

01

Closure

To show closure under multiplication, take two units in the ring, say \(a\) and \(b\). Since they are units, there exist \(a'\) and \(b'\) in \(R\) such that \(aa' = a'a = u\) and \(bb' = b'b = u\). Consider the product \(ab\), its multiplicative inverse would be \(b'a'\) because \((ab)(b'a') = a(bb')a' = aa' = u\) and \(b'a'(ab) = b'(bb')a = b'b = u\). Hence, the product \(ab\) is also a unit. Therefore, the set of units is closed under multiplication.
02

Associativity

Associativity follows directly from the fact that multiplication in a ring is associative. Therefore, if \(a\), \(b\), and \(c\) are units then \((ab)c = a(bc)\).
03

Identity

The unity \(u\) of the ring is also the identity of the group, since for any unit \(a\) in the ring, \(ua = au = a\). This means that multiplication by \(u\) doesn't change the value of a unit.
04

Inverse

The inverse of a unit \(a\) in the group is its multiplicative inverse in the ring. Since by definition of a unit, there exists an \(a'\) in \(R\) such that \(aa' = a'a = u\), it follows that each unit does have an inverse.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(p\) be a prime. (a) If \(G\) has order \(2 p\), prove that every proper subgroup of \(G\) is cyclic. (b) If \(G\) has order \(p^{2}\), prove that \(G\) has a subgroup of order \(p\).

Find the order of each element in the group of rigid motions of (a) the equilateral triangle; and, (b) the square.

Let \(S=\mathbf{R}^{*} \times \mathbf{R}\). Define the binary operation on \(S\) by \((u, v) \circ(x, y)=(u x, v x+y)\). Prove that \((S, \circ)\) is a nonabelian group.

a) In how many ways can we 5 -color the vertices of a regular hexagon that is free to move in two dimensions? b) Answer part (a) if the hexagon is free to move in three dimensions. c) Find two 5 -colorings that are equivalent for case (b) but distinct for case (a).

The following provides an alternative way to establish Lagrange's Theorem. Let \(G\) be a group of order \(n\), and let \(H\) be a subgroup of \(G\) of order \(m\). a) Define the relation \(\mathscr{\text { on }} G\) as follows: If \(a, b \in G\), then \(a \mathscr{F} b\) if \(a^{-1} b \in H\). Prove that \(\mathscr{\text { is an equivalence relation on }} G\). b) For \(a, b \in G\), prove that \(a \mathscr{b}\) if and only if \(a H=b H .\) c) If \(a \in G\), prove that \([a]\), the equivalence class of \(a\) under \(\mathscr{A}\), satisfies \([a]=a H .\) d) For any \(a \in G\), prove that \(|a H|=|H|\). e) Now establish the conclusion of Lagrange's Theorem, namely that \(|H|\) divides \(|G|\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.