Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

We prove that if a set S⊂Rn is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) [1]. On the way we show that if the ring R[ζ1,…,ζk]⊂R[X] contains the ideal (ζ1,…,ζk)R[X], then the mapping (ζ1,…,ζk):Rn→Rk is finite.