308: Ring

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A definition of ring

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Topics

About: ring

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of abelian group.
• The reader knows a definition of monoid.

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Target Context

• The reader will have a definition of ring.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

Any set, S, with the operations, addition, +, and multiplication, $\bullet$, that satisfies these conditions: 1) S is an abelian group under addition 2) S is a monoid under multiplication 3) multiplication is distributive with respect to addition, which means that $p_1\bullet (p_2+p_3)=(p_1\bullet p_2)+(p_1\bullet p_3)$ and $(p_1+p_2)\bullet p_3=(p_1\bullet p_3)+(p_2\bullet p_3)$

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2: Note

The existence of the multiplicative identity is supposed here included in being a monoid, although some people seem to define 'ring' without the requirement for the multiplicative identity; such people will call ring by this article "ring with identity".

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References

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