First, ask yourself how many groups are there of order 4.
the answer is simple => Z/4Z and Klein’s four group (K).
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 multiplicationHint 4:
to “*” then there will be 4 different rings
*upto isomorphism* right? Hence the answer is 4.
Bonus Problem:
Prove that there are only two non-ismorphic p-rings(ring with p elements) upto isomorphism.
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