On ascending chains of ideals in the polynomial ring

Abstract

Assume that $K$ is a field and $I_{1}\subsetneq ...\subsetneq I_{t}$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_{1},,...,x_{m}]$, for some $m\geq 1$. Suppose that $I_{j}$ is generated by polynomials of degrees less or equal to some natural number $f(j)\geq 1$, 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 $t\leq\mathcal{B}(m,f)$. We also discuss some possible applications of this result.