Understand the problem

The number of rings of order 4, up to isomorphism, is:
(a) 1
(b) 2
(c) 3
(d) 4.

Start with hints

Hint 1:

First, ask yourself how many groups are there of order 4.
the answer is simple => Z/4Z and Klein’s four group (K).

Hint 2 :

So intuitively there should be two rings with 4 elements.

  Okay, I will give you two rings or same order=> (R,+,.), (R,+,*) . see that order of the rings are same but I have changed the multiplication, and

I define a*b=0 for all a,b in R. [(R,+,*) is called zero ring]

Hint 3:

Question: Prove that (R,+,.) and (R,+,*) are not isomorphic! (easy)
So, we had (Z/4Z,+,.) and (K,+,.) as our answers. But if you change the multiplication 
to “*” then there will be 4 different rings 
*upto isomorphism* right?
Hence the answer is 4.

Hint 4:

Bonus Problem:
Prove that there are only two non-ismorphic p-rings(ring with p elements) upto isomorphism.

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