Math 2.0

Commutative algebra Example

The ring of integers

Builds on Commutative ring

Example

Let ℤ={…,−2,−1,0,1,2,… }\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\} be the set of integers, with the usual addition and multiplication. Then ℤ\mathbb{Z} is a commutative ring. Its zero element is 00 and its identity element is 11.

Remarks

The only units of ℤ\mathbb{Z} are 11 and −1-1. The product of two nonzero integers is not zero, thus ℤ\mathbb{Z} is an integral domain. The ring ℤ\mathbb{Z} is not a field, because 22 has no multiplicative inverse in ℤ\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionThe ring of integers, example, no evidence levelThe ring of integersexample
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. The ring of integers example

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2 pages build on this one.

Further reading

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