"Define a new addition and multiplication on $\mathbb Z$ by the rules: $a(+) b = a + b – 1$ and $a(*) b = ab – (a + b) + 2$. Prove that with these new binary operations $\mathbb Z$ is an integral domain. You may assume that under these new operations $\mathbb Z$ is a ring."
I can show that $\mathbb Z$ is a commutative ring, I'm not sure how to find the identity element of $\mathbb Z$ to show that it's an integral domain.
Thanks in advance for any help.
Best Answer
First of all, we usually call the additive identity the "zero" and the multiplicitive identity just the "identity" for clarity. I believe you are looking for zero to show that the ring is an integral domain. To do this, you need to find some $x \in \mathbb{Z}$ such that for any $a \in \mathbb{Z}$, $a(+)x = a+x-1 = a$. You can do this by algebra. A similar method can be used to find the (multiplicitive) identity.
To show that the ring is an integral domain, you need to show that, if we denote the zero element by $z$, $a\ast b = z \implies a = z $ or $b = z$.