Question

52. What is the number of ring homomorphisms from Z 12 to Z30?
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Answer 1

Answer:

Step-by-step explanation:

Let  ϕ:Z12→Z30  be an homomorphism. We know that  ϕ  is fully determined by the value of  ϕ(1) , because

ϕ(n)=n⋅ϕ(1)∀n∈Z12.  

Taking  n=30 , we get that

ϕ(30)=30⋅ϕ(1)=0.  

But, on the other hand, we have that

ϕ(30)=ϕ(6)=6⋅ϕ(1).  

From  6⋅ϕ(1)=0  we conclude that  ϕ(1)  must be a multiple of  5 . So the candidate functions are of the form  x↦5k⋅x , where  k∈{0,1,2,3,4,5} . Any such function is well defined, because

ϕ(n+12t)=5k⋅(n+12t)=5kn+60kt=5kn .

Answer 2

Answer:

may be that is helpful for you.