On ascending chains of ideals in the polynomial ring

Abstract

Assume that $K$ is a field and $I_1⊊...⊊I_t$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_1,,...,x_m]$, for some $m\ge 1$. Suppose that $I_j$ is generated by polynomials of degrees less or equal to some natural number $f(j)\ge 1$, for any $j=1,...,t$. In the paper we construct, in an elementary way, a natural number $\mathcal{B}(m,f)$ (depending on $m$ and the function $f$) such that $t\le \mathcal{B}(m,f)$. We also discuss some possible applications of this result.