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What is the general method to prove that an algebraic structure can be embedded into another?

If this is too general a question, how can I prove that $\mathbb{Z}$, or $\mathbb{Q}$ can be embedded in $\mathbb{R}$?

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By producing an embedding. – André Nicolas Jan 7 '13 at 21:33
up vote 5 down vote accepted

It depends on the structure. But in general, if you want to embed an algebraic structure $A$ into $B$. You need an injective function $f : A \to B$ which is structure preserving.

Structure preserving depends on the specific algebraic structure you are dealing with. For example, if $A$ and $B$ are groups, then $f$ needs to be a group homomorphism.

Here is a wikipedia article with more details.

For $\mathbb{Z}$ and $\mathbb{Q}$, I assume you are treating them as rings. $\mathbb{Q}$ can be embedded into $\mathbb{R}$ by the inclusion map. $f : \mathbb{Q} \to \mathbb{R}$ by $x \mapsto x$.

The properties of a ring homomorphism clearly hold.
f(x + y) = x + y = f(x) + f(y)
f(xy) = xy = f(x)f(y)
f(1) = 1

And the map is clearly injective since if $f(x) = f(y)$, then $x = f(x) = f(y) = y$.

$\mathbb{Z}$ into $\mathbb{R}$ can be done the same way.

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