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Problem 10 (Examples Of Rings Different Than The Integers)
For each item below, give an example of a ring $R$ and elements in the ring that satisfy the requested property.
- $ab=0$ but neither $a$ nor $b$ equals $0$.
- $ab=ac$ and $a\neq 0$ but $b\neq c$.
- $ab=0$ but $ba\neq 0$.
- $a^2=a$ but $a$ is neither 0 or 1.
Definition (Zero Divisor)
A zero-divisor is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b\in R$ with $ab=0$.
Definition (Integral Domain)
An integral domain is a commutative ring with unity and no zero-divisors.
Definition (Gaussian Integers)
The set $\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$ is called the Gaussian integers.
Problem 12 (The Gaussian Integers Is An Integral Domain)
Use the subring test to show that the Gaussian Integers is a subring of the complex numbers $\mathbb{C}$. Then show that the Gaussian integers is an integral domain.
For more problems, see AllProblems