Ring theory clarification

Question

Say I have a ring $R=\{a+b\sqrt{3}|~ a,~,b\in\mathbf{Z}\}$. And for all p,c,b that belong in R where p is not the additive zero. We have that p=pcb ,could I just cancel the p and say that cb=1 or do I have to say that p-pcb=0 so p(1-cb)=0 that is 1-cb=0? Also it would be great if I could get a general answer to any ring aswell. Thanks

Answer 1

In a general ring, we cannot say that ${p\ne0}$ and $p=pcb$ implies $1=cb$,

nor can we say $p\ne0$ and $p(1-cb)=0$ implies $1=cb$.

For example, in $\mathbb Z/6\mathbb Z$ (integers modulo $6$), $2\not\equiv0$ and $2\equiv2\times2\times2$ but $1\not\equiv2\times2$,

and $2(1-2\times2)\equiv0$ but $1\not\equiv2\times2$.

But in a domain (ring without non-zero zero divisors), we can make those inferences.

In particular, every subring of a field is a domain; in particular $\mathbb Z + \mathbb Z \sqrt3$ is a domain,

so for $R=\mathbb Z + \mathbb Z \sqrt3$ we can make those inferences.