Chapter 5.1, Problem 2E

Exercises

Decide whether each of the following sets is a ring with respect to the usual operations of addition and multiplication. If it is not a ring, state at least one condition in Definition 5.1a that fails to hold.

The set of all integers that are multiples of $5$.

The set of all real numbers of the form $m+n\sqrt{3}$ with $m\in \mathbb{Z}$

and $n\in \mathbb{Z}$.

The set of all real numbers of the form $a+b\sqrt[3]{5}$, where $a$

and $b$

are rational numbers.

The set of all real numbers of the form $a+b\sqrt[3]{5}+c\sqrt[3]{25}$, where $a,b$

and $c$

are rational numbers.

The set of all positive real numbers.

The set of all complex numbers of the form $m+ni$, where $m\in \mathbb{Z},n\in \mathbb{Z}$

(This set is known as the Gaussian integers.)

The set of all real numbers of the form $m+n\sqrt{2}$ with $m\in \mathbb{E}$

and $n\in \mathbb{Z}$.

The set of all real numbers of the form $m+n\sqrt{2}$ with $m\in \mathbb{Z}$

and $n\in \mathbb{E}$.