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Prove that that set of all quaternions, with the matrix addition an multiplication, is a ring with unity. Find an example to show that this ring in not commutative. You may assume matrix addition and multiplication are associative and obey the distributive law.

$\alpha = \left[ {\begin{array}{cc} a+bi & c+di \\ -c+di & a-bi \\ \end{array} } \right] $

So I need to prove that this is commutative with addition, isn't commutative with multiplication, and is associative and distributive under multiplication. It is given that addition and multiplication are associative and distributive, thus I only need to prove/disprove commutative.

Addition would be $\alpha_1+\alpha_2$ and $\alpha_2+\alpha_1$? And then multiplication would be something like taking $\alpha_1\cdot\alpha_2$ and then $\alpha_2\cdot\alpha_1$? Then I would find the multiplicative identity to prove unity.

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    $\begingroup$ So in other words, you're given that $M_{2 \times 2}(\mathbb{C})$, the set of $2 \times 2$ matrices with entries in $\mathbb{C}$ is a ring. What tool do you have to prove a subset of a known ring is also a ring? $\endgroup$ – Daniel Schepler Oct 17 '17 at 23:37
  • $\begingroup$ @DanielSchepler I did a Google search because I'm not really sure and was shown subrings? $\endgroup$ – K Math Oct 17 '17 at 23:57
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I agree with part, but not all, of what you have said.

Yes, you need to prove that $\alpha_1+\alpha_2 = \alpha_2 + \alpha_1$. Yes, commutativity of multiplication involves $\alpha_1\alpha_2$ and $\alpha_2\alpha_1$; you are supposed to disprove commutativity of multiplication, so give examples of $\alpha_1$ and $\alpha_2$ so that $\alpha_1\alpha_2 \neq \alpha_2\alpha_1$.

But those are not the only requirements to be a ring. Check that this set is closed under addition and multiplication. Show that it has additive and multiplicative identities (identify what the elements are and verify that they work). Show that each element has an additive inverse (identify what is the additive inverse of a given element and verify that it works). I'm not saying that these steps are long or difficult (or short or easy), but if your instructor expects you to verify all of the conditions to be a ring other than associativity and distributivity, then you had better be aware of what all of the conditions are.

As in @DanielSchepler's comment, some (but not all!) of these conditions follow "automatically" from the fact that your set (the quaternions) are contained in the $2 \times 2$ matrices, which you are basically given is a ring. But that is a theorem, not some instant law of mathematics. Have you proven any theorems about subrings, and do you believe your instructor would let you use such theorems for this assignment? Or are you expected to "directly" check each condition? That is a question that you might want to ask your instructor.

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  • $\begingroup$ I have done the above with your suggestions however I am not sure how to find an example of a non-commutative quaternion. I thought quaternions of complex numbers commuted because complex numbers commute. So I would need some quaternion where the complex numbers don't correspond, like $(0,i)\cdot(i,0)=(0,-1)$. $\endgroup$ – K Math Oct 18 '17 at 13:24
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    $\begingroup$ Make an example of $2 \times 2$ matrices. Like $\alpha_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, $\alpha_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$... oh, darn, those ones do commute, oopsie! You will have to think of some $\alpha_1$ and $\alpha_2$ that don't commute. $\endgroup$ – Zach Teitler Oct 18 '17 at 16:17

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