Some Remarks on the Primary ideals of $\Bbb Z_{p^m}[X]$

Commun. Korean Math. Soc. 2006 Vol. 21, No. 4, 641-652 Printed December 1, 2006

Sung Sik Woo Ewha Women's University

Abstract : In [2], they found some natural generators for the ideals of the finite ring $\Bbb Z_{p^m}[X]/(X^n-1)$, where $p$ and $n$ are relatively prime. If $p$ and $n$ are not relatively prime $X^n-1$ is not a product of basic irreducible polynomials but a product of primary polynomials. Due to this fact, to consider the ideals of $\Bbb Z_{p^m}[X]/(X^n-1)$ in `inseparable' case we need to look at the primary ideals of $\Bbb Z_{p^m}[X]$. In this paper, we find a set of generators of ideals of $\Bbb Z_{p^m}[X]/(f)$ for some primary polynomials $f$ of $\Bbb Z_{p^m}[X]$.

Keywords : primary ideal, polynomial over a finite ring