Math 2.0

statement theorem

Ring of integers

Statement

The integers Z\mathbb{Z}, with their addition ++ and multiplication ⋅\cdot, form a commutative ring with identity 11. That is, for all integers aa, bb and cc:

  1. (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) and a+b=b+aa + b = b + a;
  2. a+0=aa + 0 = a, and there is an integer −a-a with a+(−a)=0a + (-a) = 0;
  3. (ab)c=a(bc)(a b) c = a (b c) and ab=baa b = b a;
  4. a⋅1=aa \cdot 1 = a;
  5. a(b+c)=ab+aca (b + c) = a b + a c.

Remarks

The ring Z\mathbb{Z} is not a field: for example, 22 has no multiplicative inverse in Z\mathbb{Z}. The ring Z\mathbb{Z} has no zero divisors: if ab=0a b = 0, then a=0a = 0 or b=0b = 0. Thus it is an integral domain. Here "ring of integers" means the ring Z\mathbb{Z}. In algebraic number theory, the same words have a more general meaning.

Explanations

L1 human

This statement lists the usual rules of computation with whole numbers. You can add, subtract and multiply integers, and the result is always an integer. Division is different: 1/21 / 2 is not an integer.

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Integers, object, no evidence levelIntegersobjectRing of integers, statement, no evidence levelRing of integersstatement
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Sources

  1. Web https://en.wikipedia.org/wiki/Integer, Algebraic properties

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