Ring theory MOC

Integers

The integers are the initial object in Category of rings. ring Given any ring with unity , the unique ring homomorphism is given by

Proof

by basic properties of groups, and by ^P5. Note that this homomorphism is completely determined from the fact , hence it is unique.

By standard Euclidean division, forms a Euclidean domain. In some number-theoretic contexts, these are referred to as the rational integers to distinguish them from algebraic integers.

Properties

• A ring contains the integers or modular arithmetic
• A field contains modular arithmetic or the rationals
• The field of fractions of is Rational numbers

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