How to prove that an algebraic structure can be embedded into another?

Question

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 $\mathbb{R}$?

Answer 1

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. http://en.wikipedia.org/wiki/Embedding

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.