A very naturel question come in my mind but I do not know how to answer, I need a help.
For this ring $(\mathbb{Z}_2\times \mathbb{Z}_3)[X]/ (X^2)$, I was wondreing if thiere elements are like $(0,1)\overline{X}+(1,2)$ but I never see this notation!! If it is true where can I read about it?
Another question, I know that this is finite ring, so we can express all its elemnt. I think that is like $\mathbb{Z}_p[X]$ so I can use euclidian division, but here $\mathbb{Z}_2\times \mathbb{Z}_3$ is not a ring, is there a method to do that?
You claim that $(\Bbb{Z}/2\Bbb{Z})\times(\Bbb{Z}/3\Bbb{Z})$ is not a ring, but it is in fact canonically a ring. In general, the product of two rings is again a ring. In this particular case we even have the very nice canonical isomorphism $$(\Bbb{Z}/2\Bbb{Z})\times(\Bbb{Z}/3\Bbb{Z})\cong\Bbb{Z}/6\Bbb{Z},$$ so in stead of representing the coefficients of polynomials by pairs of integers, we can represent them by integers from the set $\{0,1,2,3,4,5\}$. And indeed, all elements of the quotient $$(\Bbb{Z}/6\Bbb{Z})[X]/(X^2),$$ are of the form $a\overline{X}+b$ with $a,b\in\{0,1,2,3,4,5\}$. A popular shorthand is to write $\varepsilon:=\overline{X},$ and $$(\Bbb{Z}/6\Bbb{Z})[\varepsilon]:=(\Bbb{Z}/6\Bbb{Z})[X]/(X^2),$$ in analogy with the infinitesimal $\varepsilon$ from calculus, as it satisfies $\varepsilon^2=0$.