Ascending chains of ideals in the polynomial ring

Abstract

Assume that K is a field and I1 ⊊ ... ⊊ It is an ascending chain (of length t ) of ideals in the polynomial ring K[x1, ..., xm], for some m ≥ 1. Suppose that Ij is generated by polynomials of degrees less or equal to some natural number f(j) ≥ 1, for any j = 1, ..., t. In the paper we construct, in an elementary way, a natural number B(m, f) (depending on m and the function f ) such that t ≤ B(m, f). We also discuss some applications of this result.