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It is an easy exercise to show that $\frac{\mathbb{Z}[z]}{\langle x \rangle }$ is isomorphic to $\mathbb{Z}$ using the fundamental theorem of ring homomorphisms.

I was wondering if the quotient ring $\frac{\mathbb{Z}[z]}{\langle x^2 \rangle }$ is isomorphic to a simpler ring or not?

More generally, can we simplify $\frac{\mathbb{Z}[z]}{\langle x^n \rangle }$ for $n \in \mathbb{N}$ ?

I tried attempting this using a similar approach as for $\frac{\mathbb{Z}[z]}{\langle x \rangle }$ but I am unable to find a homomorphism for which $Ker(\phi) = \langle x^n \rangle$.

Any help/hints will be highly appreciated.

J.-E. Pin
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Another_Ramanujan_Fan
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    You probably mean $\frac{\mathbb{Z}[x]}{\langle x \rangle }$ instead of $\frac{\mathbb{Z}[z]}{\langle x \rangle }$. – lhf Apr 18 '22 at 10:15

2 Answers2

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I am not sure whether it is a simpler description, but $\Bbb Z[X]/(X^n)$ is isomorphic to the ring of matrices with entries in $\Bbb Z$ of the form $$ \begin{pmatrix} a_0& a_1 & a_2 & \dotsm & a_{n-1}\\ 0 & a_0 & a_1 & \dotsm & a_{n-2} \\ 0 & 0 & a_0 &\dotsm & a_{n-3} \\ \vdots& \vdots &\vdots & \ddots & \vdots\\ 0 &0 &0 & \dotsm & a_0 \end{pmatrix} $$ For $n = 2$, this is the ring of matrices of the form $\begin{pmatrix} a&b\\ 0&a\end{pmatrix}$, with $a, b \in \Bbb Z$.

J.-E. Pin
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Its isomorphic to the ring of polynomials $\{a_0+a_1x+\ldots+a_{n-1}x^{n-1}\mid a_i\in\Bbb Z\}$ with $x^n=0$.

Wuestenfux
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