  In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by
$n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides
with the entire ring of slice regular functions. The proof required the study
of the non-commutative syzygies of a vector of regular functions, that manifest
a different character when compared with their complex counterparts.
