## The Number of Ring Homomorphisms from ℤ to ℤ N N

##### Abstract

This research deals with determining the number of homomorphism : ℤ n ℤ n , an
aspect that seems to have been left open for many years. Several researches on the
number of ring homomorphisms have been conducted and results published except for
: ℤ n ℤ n . First, we considered the number of divisors of ℤ n using the Euler’s phi
function, which are the generators of ideals. We then determined the number of ideals
in ℤ n , as all kernels of homomorphism are ideals. We determined the number of
homomorphisms : ℤ n ℤ n by finding the elements m ℤ n , such that
m mmod n 2 . A particular case of determining the number of homomorphisms m : ℤ
n ℤ n , for n 2,3,4...40 has been given, from which conclusions were drawn and
presented in form of theorems. It was deduced that the number of homomorphisms
: ℤ n ℤ n
from ℤ n to ℤ n is equal to the number of idempotent elements in ℤ n .
Finally, a generalization of the formula for finding the number of homomorphisms :
ℤ n ℤ n
from ℤ n to ℤ n is given by;
m
mtimes
k
m
k k k m n p p p ........... p 2 2 2 ......... 2 2 1 2 3
1 2 3 thus there are
2 m homomorphisms m : ℤ n ℤ n , where m p , p , p ,......., p 1 2 3 are distinct primes and
m k ,k ,k ,........, k 1 2 3 ℤ n .