An Algorithm to Compute a Primary Decomposition of Modules in Polynomial Rings over the Integers

Abstract

We present an algorithm to compute the primary decomposition of a submodule $\mathcal{N}$ of the free module $\mathbb{Z}[x_1,\dots ,x_n{]}^m$. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of $\mathcal{N}$, i.e. the minimal associated primes of the ideal $\Ann(\Z[x_1, \ldots, x_n]^m /\mathcal{N})$ in $\mathbb{Z}[x_1,\dots ,x_n]$ and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in {\sc Singular}.

10 pages

References in corpus (1)

• An algorithm for primary decomposition in polynomial rings over the integers