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# Order of rings: TIFR GS 2018 Part B Problem 12 # Understand the problem

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

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.

# Connected Program at Cheenta

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Understand the problem

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

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.

# Connected Program at Cheenta

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

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