Ring theory MOC

Integers

The integers ℤ are the initial object in Category of rings. ring Given any ring 𝑅 with unity 1, the unique ring homomorphism 𝐼 :ℤ →𝑅 is given by

𝐼:ℤ!→𝑅𝑛↦𝑛⋅1𝑅

Proof

(𝑚 +𝑛)𝑎 =𝑚𝑎 +𝑛𝑎 by basic properties of groups, and (𝑛 ⋅1)(𝑚 ⋅1) =(𝑚𝑛) ⋅1 by ^P5. Note that this homomorphism is completely determined from the fact 1 ↦1𝑅, hence it is unique.

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

Properties

• ker⁡𝐼 =(char 𝑅)ℤ
• A ring contains the integers or modular arithmetic
• A field contains modular arithmetic or the rationals
• The field of fractions of ℤ is Rational numbers

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