(This is not meant to be too rigorous, so please bear with me)

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For example, if you add, subtract, or multiply two integers, you get another integer. So the integers are a ring.

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On the other hand, the rational numbers are closed under division too, which makes them a field.

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So it's now interesting to see which ones can be divided by which!

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So rings are not just numbers, functions can also form a ring. You can add, subtract, and multiply them.

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Or, put another way, you know the ring, you know the shape.

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This is one motivation for the subject of algebraic geometry.

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See, ideals were originally invented for number theory. But again, rings are not just numbers!

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You know the ring, you know the shape.

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They are a ring after all, and determine a shape.

You know the ring, you know the shape. At least we should...

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