My Math Forum field-ring

 Abstract Algebra Abstract Algebra Math Forum

 June 30th, 2017, 05:18 AM #1 Member   Joined: Nov 2011 Posts: 73 Thanks: 0 field-ring 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 non-zero 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
Quote:
 Originally Posted by skipjack In a field, the non-zero elements form a commutative group (under "multiplication").

July 1st, 2017, 10:17 AM   #4
Senior Member

Joined: Aug 2012

Posts: 1,527
Thanks: 364

Quote:
 Originally Posted by MATHEMATICIAN then what about ring?
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.

 Tags fieldring

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post shaharhada Abstract Algebra 2 October 23rd, 2016 03:35 AM shaharhada Linear Algebra 3 July 18th, 2015 07:50 AM watson Abstract Algebra 1 September 14th, 2012 09:07 PM tinynerdi Abstract Algebra 5 May 16th, 2010 11:42 AM xixi Abstract Algebra 0 January 30th, 2010 08:21 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top