$R=\mathbb{Z}[x]$ - polynomials with integer coefficients
$I = (x^2+x-1)\mathbb{Z}[x]$
Am i right that classes of factor-ring $R/I$ represented in the next form: $K_{a,b} = ax+b$?
Also I have question: there is no zero divisors in the ring $R$ and factor-ring $R/I$?
I highly appreciate all answers. Thank you in advance.
Indeed, since $I$ is generated by a MONIC polynomial of degree $2$, every equivalence class of the quotient ring $R/I$ can be represented in the form of a linear polynomial over $\mathbb{Z}[x]$. This is mainly because $\mathbb{Z}[x]$ is a domain (in fact, UFD), and since the leading coefficient of $x^{2}+x-1$ is a unit, one can perform polynomial division by $x^{2}+x-1$ to obtain an element of the class with degree strictly less than $2$.
Your second question is somewhat related to your first. Since $\mathbb{Z}$ is a UFD, it follows that $\mathbb{Z}[x]$ is also a UFD, hence a domain. This is a famous theorem of Gauss's - it is sometimes called Gauss's lemma. You may want to see these wonderful notes for more information. Note that $x^{2}+x-1$ is irreducible over $\mathbb{Z}[x]$ (why? You might appeal to any number of classical results here, including the rational root theorem, though there are more elementary methods). Hence, $I$ is a prime ideal, and therefore $R/I$ is also a domain.