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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Chapter
Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Label each of the following statements as either true or false

The characteristic of a ring R is the smallest positive integer n such that n x = 0 for

some x in R

To determine

Whether the statement, “The characteristic of ring R is the smallest positive integer n such that nx=0 for some x in R ” is true or false.

Explanation

Formula used:

Definition of characteristic of a ring:

If there are positive integers such that nx=0 for all x in R, then the smallest positive integer m such that mx=0 for all x in R is called the characteristic of ring R. If no such positive integer exists, then R is said to be of characteristic zero.

Explanation:

Consider the statement, “The characteristic of ring R is the smallest positive integer n such that nx=0 for some x in R.”

Counter example:

Let 6={[0],[1],[2],[3],[4],[5]}

[ab]=[a][b]

Multiplication table for non-zero element in 6 is

[1][2][3][4][5][1][1][2][3][4][5][2][2][4][0][2][4][3][3][0][3][0][3][4][4][2][0][4][2][5][5][4][3][2][1]

From the above table,

3[0]=[30]=[0]

3&#

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