Are you sure the question is whether there is a homomorphism between and and not just between C and ?
There is an obvious homomorphism .
is there a homomorphism
Φ:C×C→M₂(R)
Φ must be onto
I have tried variations on the type of entries in the 2 by 2 matrices that will preserve the additive and multiplicative properties of a homomorphism but this seems quite long. Is there a better way for me to find if a homomorphism exists that is onto?
i still cant figure this out. I see that there are 4 entries of real numbers but how can i use this fact to get a surgective homomorphism?
the only thing i can think of is that 4 linearly independant 2*2 matrices will generate "all" real 2*2 matrices. But i dont think the the multiplication property of an isomorphism holds in the case where i take the matrix entries to be a,b,c,d.
but does the homomorphism have to be comutative?
When i chech the multiplication property F{(x,y)(u,v)}=F(x,y)F(u,v)
Do i multiply components only or everything?
eg
F{(a+bi,c+di)(e+fi,g+hi)}
doi have
F{(ae-fi+afi+bei,cg-dh+chi+dgi)}
or do i multiply everything?
Because in this case a homorphism does exist even though it is not surjective, otherwise no homomorphism exists.
In general the homomorphic image of a commutative ring (obviously a ring hom.) is commutative: Take where is commutative and is simply a ring, then without loss of generality is onto (because is a subring) then take then for some then .
I'm assuming of course that you're giving the usual structure of a product ring (or some commutative structure at least)
the question specifcally says:is there ANY homomrphism, so i'm presuming he wants us to think about this question.
I have basically showed 1 example that is a ring homomorphism but is not surjective, and a map that is surjective but does not preserve the ring homomorphism properties.And additionally stated that C*C is commutative but matrices aren't.
Does a homomrphism exist though that is NOT necassarily a ring homomorphism?Because i have answered this as no, because i cant find any homomorphism that is surjective.