Chapter 6.2, Problem 30E

In the ring $\mathbb{Z}$ of integers, let new operations of addition and multiplication be defined by

$x\oplus y=x+y+1$ and $x\odot y=xy+x+y,$

Where $x$ and $y$ are arbitrary integers and $x+y$ and $xy$ denote the usual addition and multiplication in $\mathbb{Z}$.

Prove that the integers from a ring $R^{\text{'}}$ with respect to $\oplus$ and $\odot$.

Identify the zero element and unity of $R^{\text{'}}$.

Prove that $\mathbb{Z}$ is isomorphic to $R^{\text{'}}$.