Ideal of the ring
Let's assume that is a ring, which operations are addition and multiplication using components (eg. ). The task is to prove that x-axis and y-axis are ideals of this ring. Following theorem must be used in the proof:
"Ring homomorphism's kernel is 's ideal".
I think the problem isn't very hard, when the correct homomorphism is found. I tried some, but I couldn't discover anything that works. So, any help is welcome. Thank you very much!
Well, the -axis is, by definition, the set
If you want to prove that is an ideal using the statement you have written, we need a ring and a ring homomorphism such that .
How about taking , and defining
(projection onto the first coordinate)? It should be fairly simple to prove that is indeed a ring homomorphism, and that the kernel of is exactly the set of points located on the -axis.
You can do a completely analogous thing to show that the axis is also an ideal.