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College Mathematics

Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.

Understand the problem

The number of rings of order 4, up to isomorphism, is:
(a) 1
(b) 2
(c) 3
(d) 4.
Source of the problem
TIFR GS 2018 Part B Problem 12
Topic
Abstract Algebra
Difficulty Level
Easy
Suggested Book
Dummit and Foote

Start with hints

Do you really need a hint? Try it first!

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]

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

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