June 30th, 2017, 05:18 AM  #1 
Member Joined: Nov 2011 Posts: 73 Thanks: 0  fieldring
What is the difference between a ring and a field?
Last edited by skipjack; June 30th, 2017 at 05:49 AM. 
June 30th, 2017, 05:51 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1385 
In a field, the nonzero elements form a commutative group (under "multiplication").

July 1st, 2017, 09:36 AM  #3 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 831 Thanks: 60 Math Focus: सामान्य गणित  
July 1st, 2017, 10:17 AM  #4 
Senior Member Joined: Aug 2012 Posts: 1,527 Thanks: 364  Some elements of a ring don't have multiplicative inverses. The classic example of a ring is the integers. You can add, subtract, and multiply; but you can't always divide. For example $2$ has no multiplicative inverse. But in the rationals, every nonzero element has a multiplicative inverse. For example the inverse of $2$ is $\frac{1}{2}$. To further clarify the distinction between rings and fields, we can consider the finite case. The integers mod $5$ are a field. You can convince yourself that any nonzero element has a multiplicative inverse. For example the inverse of $2$ is $3$, because $2 \times 3 = 6 \equiv 1 \pmod 5$. The integers mod $6$ are a ring (you can add, subtract, and multiply) but not a field. For example $2$ has no multiplicative inverse. Last edited by Maschke; July 1st, 2017 at 10:22 AM. 

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