   Chapter 6.3, Problem 2TFE

Chapter
Section
Textbook Problem

Label each of the following statements as either true or falseThe characteristic of a ring R is the smallest positive integer n such that n x = 0 forsome 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={,,,,,}

[ab]=[a][b]

Multiplication table for non-zero element in 6 is



From the above table,

3==

3&#

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