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August 13th, 2016, 06:52 AM   #1
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ring vs. field

What is the difference between a ring and a field?

Last edited by skipjack; October 23rd, 2016 at 04:49 AM.
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August 13th, 2016, 08:56 AM   #2
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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.
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October 23rd, 2016, 03:35 AM   #3
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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).
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Last edited by skipjack; October 23rd, 2016 at 04:55 AM.
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