The ModularArithmetics? are the images of the IntegerNumbers under group/ring HomoMorphism*s*. Such an operation is going to zero out some NormalSubgroup/Ideal?, and these turn out to be precisely the sets of the form pZ for some integer p; the resulting group/ring is denoted Zp.

To put it another way, Zp consists of the remainders {0,1,...,p-1}, so that p=0. For instance, Z3 has the following addition and multiplication tables:

0+0=0 1+0=1 2+0=2 0+1=1 1+1=2 2+1=0 0+2=2 1+2=0 2+2=1

0*0=0 1*0=0 2*0=0 0*1=0 1*1=1 2*1=2 0*2=0 1*2=2 2*2=1

When p is a composite number, the factors of p are going to turn out to be ZeroDivisors?. When p is prime, these don't exist, and so Zp is an IntegralDomain? and in fact necessarily a field.