# $(\mathbb{Z}_2\times \mathbb{Z}_3)[X]/ (X^2)$

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?

• $Z_2xZ_3 \cong Z_6$ Commented May 1, 2019 at 21:37
• thanks but this do not answer my question, I want to know this ring not what it is isomorphic to Commented May 1, 2019 at 21:40

## 1 Answer

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 $$\varepsilon^2=0$$.