Additive Group of Integers is Subgroup of Rationals

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Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\Q, +}$ be the additive group of rational numbers.


Then $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.


Proof

Recall that Integers form Integral Domain.

The set $\Q$ of rational numbers is defined as the quotient field of the integers.

The fact that the integers are a subgroup of the rationals follows from the work done in proving the Existence of Quotient Field from an integral domain.

$\blacksquare$


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