Math 2.0

example

The ring of integers

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}.

Explanations

L1 human

The integers are the first example of a ring. You can add, subtract and multiply two integers, and the result is an integer. You cannot always divide: 1/21 / 2 is not an integer.

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Sources

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

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