Is this correct anyone;
C is the group of all complex numbers under addition
O : C ----> C
z ----> z + iz
For all z1,z2 E C
O (z1+z2) = z1 + z2 + iz1 + iz2 = z1+iz1 + z2+iz2 = O (z1) + O (z2)
Hence O satisfies the homomorphism property and so is a homomorphism.
explicitly, let be given, and define:
by
then
this is what distributivity tells us in a commutative ring R: that multiplication by a fixed element r is an additive endomorphism. a slightly more archaic way of saying this, is that multiplication is compatible with addition.
I am trying to find the image and kernal of this homomorphism, am I correct in saying
The identity element in (C, +) is 0, so
Ker O = {z E C : O (z) =0}
= (z E C : z+iz = 0}
= {0}
Because z + iz = z for each z E C, the function O is onto and
Im (O) = C.
saying what you wish to be true, is not what constitutes a "proof" or demonstration.
in particular, if you claim that ker(O) = {0}, you have to give a reason for it.
one possible justification is that O(z) = z+iz = z(1+i). so if O(z) = z(1+i) = 0,
then z(1+i)((1-i)/2)) = 0((1-i)/2) --> z = 0. this shows that ker(O) is a subset of {0},
and obviously we have 0 is an element of ker(O), so ker(O) = {0}.
now, your argument that O is onto is just not one at all.
in the first place, z+ iz DOES NOT EQUAL z (unless z = 0).
to prove O is onto, given any w in C, you have to FIND some z in C with O(z) = w.
to do this, you first set w = z + zi = z(1+i), and try to solve for z.
then z = w/(1+i).
does this work? well: O(w/(1+i)) = (w/(1+i)) + (w/(1+i))i = (w/(1+i))(1+i) = w((1+i)/(1+i)) = w(1) = w.
THAT shows O is onto, so that Im(O) = C.
one could do many things.
for example, one could use the fact that since C is a field, it is a fortiori, an integral domain, so:
z(1+i) = 0, implies one of z or 1+i is 0. since 1+i is not 0, z must be.
dividing by 1+i is the same as multiplying by 1/(1+i).
since (1+i)((1-i)/2)) = (1/2)(1+i)(1-i) = (1/2)(1^{2} - i^{2}) = (1/2)(1 - (-1)) = (1/2)(1 + 1) = (1/2)(2) = 1,
we see that 1/(1+i) *IS* (1-i)/2. in other words, "dividing by 1+i" is exactly what i did.