My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Thanks Tree2Thanks
  • 1 Post By IAmABread
  • 1 Post By Maschke
Reply
 
LinkBack Thread Tools Display Modes
April 28th, 2017, 02:18 PM   #1
Newbie
 
Joined: Apr 2017
From: Europe

Posts: 16
Thanks: 1

Rings with four elements

Good morning.
Recently I've been doing some research on a ring theory. I found out that there 11 rings that have four elements (is the number of them correct or are they less or more?)? Here comes my question: why are there only 11 of them? Can someone explain this in a language of abstract algebra?
Thanks from agentredlum
IAmABread is offline  
 
April 28th, 2017, 02:42 PM   #2
Senior Member
 
Joined: Aug 2012

Posts: 1,521
Thanks: 364

Is that right? Great factoid. I've played with the field with four elements but I never thought about this question for rings. Are these commutative? With or without identity?

ps -- Today I learned! These ring are not necessarily commutative, and need not contain a multiplicative identity.

It's a fact that there are exactly 11 such rings of order $p^2$ if $p$ is a prime.

I found a nice Stackexchange thread. https://math.stackexchange.com/quest...ith-4-elements.

One answer refers to a program alg which lets you input a bunch of axioms and cranks out all the models of the axioms. If only Russell and Hilbert had software like this!

I also found this paper that proves the result that there are 11 rings of order $p^2$.

I'm afraid I'm completely unable to shed the slightest bit of light on this beyond Googling.
Thanks from agentredlum

Last edited by Maschke; April 28th, 2017 at 02:53 PM.
Maschke is offline  
April 28th, 2017, 03:37 PM   #3
Newbie
 
Joined: Apr 2017
From: Europe

Posts: 16
Thanks: 1

Well, since you have googled the topic, it seems like I don't need to explain anything. :P But I did the same when I came up with my 11 rings, but I wasn't able to find the paper you mentioned at the end. It is really interesting. Thank you kindly, but I'd like to invite others to discuss the problem. Maybe it is not the only way to justify this. Source of my knowledge limits me a bit, because I can say I started dealing with ring theory not long time ago.
IAmABread is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
elements, rings



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Rings ewilfredo91 Abstract Algebra 5 September 28th, 2014 01:02 PM
Rings. Z4[x] math interest22 Abstract Algebra 5 February 5th, 2014 06:30 AM
Rings alejandrofigueroa Abstract Algebra 1 November 17th, 2013 05:26 PM
Nilpotent elements in rings PeterMarshall Abstract Algebra 2 January 20th, 2012 01:15 AM
Rings bjh5138 Abstract Algebra 0 December 4th, 2007 03:22 PM





Copyright © 2017 My Math Forum. All rights reserved.