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 N of the free module Z [x1, . . . , xn] 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 N , i.e. the minimal associated primes of the ideal Ann . . , xn] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of ShimoyamaYokoyama. The algorithms are implemented in Singular.

F. S. Macaulay: From plane curves to Gorenstein rings

“…This must have been a concern; Emmy Noether famously asked whether the decomposition could be effected at all algorithmically ("in endlich vielen Schritten"), a question that her student Grete Hermann resolved positively in characteristic 0. (There are now many such algorithms, in all characteristics and even over the integers; see for example [IPS15] and the references there.) 6.1.…”