Question

How many homomorphisms are there of z into z?

 

Answer 1

Are we talking group homomorphisms or ring homomorphism from the integers to themselves? The integers as a group is cyclic and generated by 1. This means that any homomorphism f into itself is determined by the image of 1, say f(1)=a . In additive notation

f(n)=f(1+⋯+1)=f(1)+⋯+f(1)=nf(1)=na .

There is clearly one of those for each integer a, yielding a countable number of group homomorphisms.

If f is a homomorphism of rings is also a homomorphism of additive groups so all of the above applies. Then f also obeys f(mn)=f(m)f(n) . In particular f(1)=a but also f(1)=f(1²)=f(1)f(1)=a² . This means that a is either 0 or 1, since a2=a by equating the two. Thus there are exactly two ring homomorphisms. (If ring homomorphisms are assumed to preserve the multiplicative identity then we only get one of these allowed.)

 There are exactly two ring homomorphisms from Z to Z. (If ring homomorphisms are assumed to preserve the multiplicative identity then we only get one of these allowed.)