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.


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 $\mathbb{Z}_{r}$.

  • $\begingroup$ 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 $r_1, r_2, \cdots, r_t\in \mathbb{N}$ ? $\endgroup$ – Anjan3 Dec 3 '15 at 6:04
  • $\begingroup$ I will let you think about it! (Notice that the quotient ring you gave is a vector space over $\mathbb{Z}_{p}$) $\endgroup$ – mich95 Dec 3 '15 at 6:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.