Let $(R,+,\times)$ be a ring with additive identity $0 \in R$. On the set $S = \{(a,b) :\ a,b \in R\}$ the binary operators $\oplus$ and $\otimes$ are defined by:
$$(a, b)\oplus(c, d) = (a+c, b+d),$$ $$ (a, b)\otimes(c, d) = (a\times c+b\times d, a\times d+b\times c).$$
I am currently doing self study of abstract algebra and need some guidance. I need to state a non-trivial mapping $\theta : S \to R$ and show that it is a homomorphism.
Firstly I done (& please correct me if I am wrong):
let $θ : S \to R$ defined by $\theta(a,b) = a$.
Then
$$\theta((a,b) + (c,d)) = (a,b) + (c,d) = (a+c, b+d)$$ $$\theta((a,b)) + θ((c,d)) = (a,b) + (c,d) = (a+c, b+d)$$
$$\theta((a,b) \times (c,d)) = (a,b) \times a (c,d) = (ac + bd, ad + bc)$$ $$\theta((a,b)) \times θ((c,d)) = (a,b) \times (c,d) = (ac+ bd, ad + bc)$$
Therefore it is a homomorphism
Does this then give me what I am looking for? Please feel free to state anything that I have done incorrectly.
Thank you