# Prove or disprove ring $\mathbb{C}\times \mathbb{C}$ and ring Quaternion $H$ are isomorphic

My attempt: let $$f$$ ($$w$$+$$x$$i+$$y$$j+$$z$$k) $$=$$ ($$w$$+$$x$$i,$$y$$+$$z$$i)

then I tried proving it is a homomorphism. $$f$$ is a homomorphism under addition but fails to be a homomorphism under multiplication.

Can anyone give any hint? Or any property I need to show they don't share.

Any help would be appreciated.

• What multiplication did you define on $\mathbb{C} \times \mathbb{C}$? – Harnak Feb 3 at 10:14
• @Harnak I'm not sure. I used $(a, b)(c, d) = (ac, bd）$ for $a,b,c,d$ in $C$ – Flashhh Feb 3 at 10:18
• You are trying to prove that they are isomorphic with respect to which structure? – José Carlos Santos Feb 3 at 10:21
• @JoséCarlosSantos They are both rings, I have edited it. – Flashhh Feb 3 at 10:22
• Then you did you use the group-isomorphism tag? – José Carlos Santos Feb 3 at 10:24

They are not isomorphic because $$\mathbb{C}\times\mathbb{C}$$ is commutative, whereas $$\mathbb H$$ is not.
• How did you define multiplication on $\mathbb{C} \times \mathbb{C}$ ? Sorry I'm new to this topic. – Flashhh Feb 3 at 10:30
• I used the one that you mentioned in the comments: $(a,b)\times(c,d)=(ac,bd)$. – José Carlos Santos Feb 3 at 10:32
To have a ring isomorphism you need to give $$\mathbb{C}^2$$ the following ring structure: $$(a,b)+(c,d) = (a+c,b+d)$$ $$(a,b)(c,d) = (ac - \bar{d}b, da + b \bar{c})$$ in which case the isomorphism is given by: $$x + yi + zj + wk \mapsto (x + yi, z + wi)$$ It doesn't work with the product you mentioned in the comments above.
• That depends on what you have to do. There is no standard product in $\mathbb{C}^2$. If you want them to be ring-isomorphic, however, you have to define the product I mentioned. – Harnak Feb 3 at 10:46