Integers

The integers $ℤ$ are the initial object in $𝖱 𝗂 𝗇 𝗀$ . #m/thm/ring Given any ring $𝑅$ with unity $1$ , the unique ring homomorphism $𝐼 : ℤ \to 𝑅$ is given by

𝐼 : ℤ! \to 𝑅 𝑛 \mapsto 𝑛 \cdot 1 𝑅

Proof

$(𝑚 + 𝑛) 𝑎 = 𝑚 𝑎 + 𝑛 𝑎$ by basic properties of groups, and $(𝑛 \cdot 1) (𝑚 \cdot 1) = (𝑚 𝑛) \cdot 1$ by ^P5. Note that this homomorphism is completely determined from the fact $1 \mapsto 1 𝑅$ , 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

$k e r 𝐼 = (c h a r 𝑅) ℤ$
A ring contains $ℤ$ or $ℤ 𝑛$
A field contains $ℤ 𝑝$ or $ℚ$
The field of fractions of $ℤ$ is $ℚ$

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