Example

We take R = Z/2Z and f = X2 + X + 1. There are 4 elements in R[X]/(f): 0, 1, a = X and a + 1. Here is the multiplication table for these elements.

·
   0  
 1
a
a + 1
   0  
   0  
   0  
   0  
   0  
 1
   0  
 1
a
 a + 1 
a
   0  
a
 a + 1 
 1
 a + 1 
   0  
 a + 1 
 1
a

The table shows that a and a + 1 are each other's inverses.

Compare this table with the multiplication table of Z/4Z. In Z/4Z there is no element b with 2b = 1. The element 2 of Z/4Z has no inverse. Therefore, the arithmetical system on 4 elements we have just constructed is fundamentally different from Z/4Z.