Some examples on rings

Give an example on the following:

1.  A commutative ring that is not a field but its characteristic is a prime number.
2. An example of a ring that has infinite elements but a finite characteristic.
3. An integral domain that has a finite characteristic.

Solution

Here are some answers.

1. A commutative ring that is not a field but has as a characteristic a prime number is:
   
   $$\mathbb{Z}_2 \times \mathbb{Z}_2$$
   
   The characteristic is ${\rm char}\left( \mathbb{Z}_2 \times \mathbb{Z}_2\right)= {\rm lcm} (2, 2)=2$. 


2. Using the first question we see that a ring that has infinite elements but a finite characteristic is the following:
   
   $$\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots$$


3. One example could be $\mathbb{Z}_2[x]$. Indeed, the characteristic is $2$ (fair and square) and it is an integral domain since if we multiply two polynomials of this rings we never get the zero one. Thus, $\mathbb{Z}_2[x]$ has no zero divisors. So, yes it is an integral domain.