I have a question which I simply do not understand and I've been researching it quite a lot.
So I have this problem,
$(H,+,\cdot)$ where $H=\{x\in \Bbb R|x=m+n \sqrt5\ where\ m,n\in \Bbb Z\}$ and + and $\cdot$ are the sum and product respectively.
Find all zero divisors and say if its an integral domain.
Edit: Yes! $m,n \in \Bbb Z$
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So I understand that a number is a zero divisors if $ab=0$ but $a \land b \neq 0$, But does that include the only the second operation in $(H,+,\cdot)$? I simply do not understand how to tackle this. Thanks again!
$H=\{x\in \Bbb R|x=m+n \sqrt5\ \ \text{ where } \ m,n\in \Bbb N \}$ is not a ring, because it does not have a zero (there is no element $z \in H$ such that $\forall h \in h h+z=h$).
I think you may have made a typographical error an intended to type $m,n\in \Bbb Z$ or $m,n\in \Bbb Q$.
Let's do $H=\{x\in \Bbb R|x=m+n \sqrt5\ \ \text{ where } \ m,n\in \Bbb Z \}$. We want to see if there are any zero divisors in the ring; that is, we want to see if there are any $\alpha, \beta \in H$ such that $\alpha \cdot \beta=0$ but neither of $\alpha, \beta$ is zero.
But $H \subset \mathbb{R}$, so if there were two nonzero elements of $H$ whose product was zero, there would be two real numbers, $\alpha, \beta$ such that $\alpha \cdot \beta = 0$. Is this possible?
Now, we have to determine if $H$ is an integral domain. We have to check $3$ propreties:
$1)$ Does $H$ contain a $1$?
$2)$ Is $H$ commutative?
$3)$ Does $H$ have no zero divisors?
If the answers to these questions are Yes Yes No, then $H$ is an integral domain. Otherwise, it is not an integral domain.