Chapter 3.2, Problem 3E

Let R be the equivalence relation on $\mathbb{Z}\times \left(\mathbb{Z}-\left\{0\right\}\right)$ given by $\left(x,y\right)R\left(u,v\right)$ if $xv,yu.$ Let P be the set of equivalence classes of $\mathbb{Z}\times \left(\mathbb{Z}-\left\{0\right\}\right)$ modulo R. For $\left(a,b\right)$ and $\left(c,d\right)$ in P, define the operations $\oplus$ and $\otimes$ by $\left(a,b\right)\oplus \left(c,d\right)=\left(ad+bc,bd\right)$ and $\left(a,b\right)\otimes \left(c,d\right)=\left(ac,bd\right).$

1. Find the additive identity in this ring, the unity element, and the additive and multiplicative inverses of $\left(2,5\right).$ Hint: For each answer, you must give a representative of the equivalence class in P.
2. Suppose that $f:\left(\mathbb{Q},+,\cdot \right)\to \left(P,\oplus ,\otimes \right)$ is given by $f\left(p/q\right)=\left(p,q\right).$ Prove that fis a ring homomorphism.