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 August 13th, 2016, 06:52 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 ring vs. field What is the difference between a ring and a field? Last edited by skipjack; October 23rd, 2016 at 04:49 AM. August 13th, 2016, 08:56 AM #2 Senior Member   Joined: Aug 2012 Posts: 2,357 Thanks: 740 In a field, you can divide by any non-zero element. For example, in the integers we can not divide 5 by 2. The integers are a ring but not a field. In the rationals, we can divide 5 by 2. Technically, we express this by saying that in a field, every non-zero element has a multiplicative inverse. Last edited by skipjack; October 23rd, 2016 at 04:56 AM. October 23rd, 2016, 03:35 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 In a field, every member, except the additive identity, has a multiplicative inverse. The set of all rational numbers and the set of all real numbers, with the usual addition and multiplication, are fields. The set of all integers, "modulo a prime number" is also a field. For example, the integers modulo 5: the multiplicative inverse of 1 is 1 (1x1= 1), of 2 is 3, of 3 is 2 (2x3= 6= 1 (mod 5), and the multiplicative inverse of 4 is 4 (4x4= 16= 1 (mod 5). A ring satisfies the same rules as a field except that there are non-zero members that do not have multiplicative inverses. The set of all integers is an example of a ring that is not also a field. The set of integers "modulo a composite number" is a ring, not a field. For example, the set of integers modulo 6 is a ring not a field, since 2 has no inverse: 2x1= 2. 2x2= 4, 2x3= 6= 0 (mod 6), 2x4= 8= 2 (mod 6), and 2x5= 10= 4 (mod 6). Thanks from beesee Last edited by skipjack; October 23rd, 2016 at 04:55 AM. Tags field, ring Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shaharhada Linear Algebra 3 July 18th, 2015 07:50 AM rayman Abstract Algebra 1 March 11th, 2014 03:27 PM 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

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