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 June 30th, 2017, 06:18 AM #1 Senior Member   Joined: Nov 2011 Posts: 173 Thanks: 2 field-ring What is the difference between a ring and a field? Last edited by skipjack; June 30th, 2017 at 06:49 AM.
 June 30th, 2017, 06:51 AM #2 Global Moderator   Joined: Dec 2006 Posts: 18,442 Thanks: 1462 In a field, the non-zero elements form a commutative group (under "multiplication").
July 1st, 2017, 10:36 AM   #3
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Quote:
 Originally Posted by skipjack In a field, the non-zero elements form a commutative group (under "multiplication").
then what about ring?

July 1st, 2017, 11:17 AM   #4
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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 11:22 AM.

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