# Will be the ring $\mathbb{Z}_p [x]/\langle \big(f(x)\big)^m \rangle$ ring isomorphic to the ring $\mathbb{Z}_r$ for some $r$?

Suppose that we have the ring $\mathbb{Z}_p [x]/\langle \big(f(x)\big)^m \rangle$ where $p$ is a prime, $m\in \mathbb{N}, f(x)$ is irreducible polynomial over the field $\mathbb{Z}_p$. In this paper it has been shown how to compute the group of units. Today I came to know about it and while I was studying it, the following question in my mind.

In case, if we were asked to find ring isomorphism class of the ring $\mathbb{Z}_p [x]/\langle \big(f(x)\big)^m \rangle$, can we find such $r$ that the ring $\mathbb{Z}_p [x]/\langle \big(f(x)\big)^m \rangle$ will be ring isomorphic to the ring $\mathbb{Z}_r$ ?

No idea how to crack this result. If it has been done in any research paper, can some one please help me to get the link ? Or can we solve it ?

Thanks in advance.

P.S. If you find my question needs to be edited, please feel free to do so to make it more appropriate question.

## 1 Answer

Any ring homomorphism, will necessarly induce an abelian group homomorphism with respec to addition. $\mathbb{Z}_{p}[x]/(x^{2})$ is isomorphic as an abelian group to $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$, which is not isomorphic to any $\mathbb{Z}_{r}$.

• Nice answer. So in that case, should we suppose that the ring will be isomorphic to $\mathbb{Z}_{r_1}\times \mathbb{Z}_{r_2}\times \cdots \times \mathbb{Z}_{r_s}$ for some $r_1, r_2, \cdots, r_t\in \mathbb{N}$ ? – Anjan3 Dec 3 '15 at 6:04
• I will let you think about it! (Notice that the quotient ring you gave is a vector space over $\mathbb{Z}_{p}$) – mich95 Dec 3 '15 at 6:07