Gröbner bases of modules over reduction rings (Q687617)

By a reduction ring the author means a ring satisfying some axioms in order to be able to compute Gröbner bases with the Buchberger algorithm. The notion, already considered by several authors, is quite general and allows the definition of Gröbner bases in a large class of rings. In the paper it is showed that, if \(R\) is a reduction ring, then the ring \(R^ n\) (where addition and multiplication are defined componentwise) is also a reduction ring (except for one of the axioms which, however, does not compromise the application of Buchberger algorithm). Successively, it is proved that if \(F \subseteq R^ n\) is any finite subset, then a Gröbner basis of the \(R\)-module generated by \(F\) can be computed from the Buchberger algorithm applied considering \(F\) as a set of elements of the reduction ring \(R^ n\). This allows to define Gröbner bases of modules over a reduction ring, using only the notion of Gröbner bases for ideals.