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June 30th, 2017, 05:18 AM   #1
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field-ring

What is the difference between a ring and a field?

Last edited by skipjack; June 30th, 2017 at 05:49 AM.
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June 30th, 2017, 05:51 AM   #2
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In a field, the non-zero elements form a commutative group (under "multiplication").
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July 1st, 2017, 09:36 AM   #3
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Originally Posted by skipjack View Post
In a field, the non-zero elements form a commutative group (under "multiplication").
then what about ring?
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July 1st, 2017, 10:17 AM   #4
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Originally Posted by MATHEMATICIAN View Post
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.
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