Rings Isomorphism: Proving R & R_2 Subrings of Z & M_2(Z)

[dreamer.ande]

3. Let $$R = a+b \sqrt{2}$$ , a,b is integer and let $$R_{2}$$ consist of all 2 x 2
matrices of the form $$[\begin{array}{cc} a & 2b \\ b & a \\ \end{array} }]$$

Show that R is a subring of $$Z(integer)$$ and $$R_{2}$$ is a subring of $$M_{2} (Z)$$. Also. Prove that the mapping from R to $$R_{2}$$ is a isomorphism.

 

[micromass]

Hi, what did you try already and where are you stuck? Then we'll know how to help you!

 

[dreamer.ande]

I am lost totally in this question. I know i need to do this.
need to show it is closed under addition, multiplication containing identity to prove it is a subring and show that it is surjective and injective. to show that it is isomorphism.

Can you help me out in this question?

 

[micromass]

So, closed under addition/multiplication, what does that mean?? How would you show this?

 

[dreamer.ande]

that the part i am struck with. $$a+b\sqrt{2} + c+d\sqrt{2}$$ ? but $$\sqrt{2} not an integer$$

 

[micromass]

that the part i am struck with. $$a+b\sqrt{2} + c+d\sqrt{2}$$ ? but $$\sqrt{2} not an integer$$

You will want to write $$a+b\sqrt{2}+c+d\sqrt{2}$$ in the form $$e+f\sqrt{2}$$, since by definition, elements in R have this form.

We are not claiming that $$\sqrt{2}$$ is an integer!

 

[dreamer.ande]

how many axiom do i need to show to prove that it is a subring of a ring?
There are five axiom to show?
How to show that first axiom : containment. R belong to Z?

Can you show me the proper step of proving in such question as I always have problem in such question?

 

[micromass]

how many axiom do i need to show to prove that it is a subring of a ring?

Normally all of them. But luckily you have theorems that limit the amount of work you need to do.

A certain theorem tells you that you only need to show that R is closed under addition and multiplication, that 0 and 1 are in R, and that -a is in R for every a.

So you only need to show 5 things. However, if you did not see such a theorem, then you will need to show all the axioms!

 

[dreamer.ande]

Could you show me how to do it for the first part and I try out the second part?
Sorry, I am new to this thing. Very Confusing for me.