Chapter 5.4, Problem 1TFE

True or False

Label each of the following statements as either true or false.

The field $Q$ of rational numbers is an extension of the integral domain $Z$ of integers.

To determine

Whether the statement, “Every integral domain contains set of positive elements” is true or false.

Answer to Problem 1TFE

Solution:

The statement, “Every integral domain contains set of positive elements” is false.

Explanation of Solution

Formula used:

Definition of ordered integral domain:

If $D$ is an integral domain, then for each $x\in D$, one and only one of the following statements is true:

$x\in D^{+},x=0,-x\in D^{+}$.

Explanation:

Consider the statement, “Every integral domain contains set of positive elements.”

By using definition, an element $x\in D$ such that $-x\in D^{+}$ is called a negative element of $D$.

Thus, every integral domain may or may not contain the set of positive elements.

Hence, the statement “Every integral domain contains set of positive elements” is false.