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Assume that $K$ is a field and $I_{1}\subsetneq ...\subsetneq I_{t}$ is anascending 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 bypolynomials of degrees less or equal to some natural number $f(j)\geq 1$, forany $j=1,...,t$. In the paper we construct, in an elementary way, a naturalnumber $\mathcal{B}(m,f)$ (depending on $m$ and the function $f$) such that$t\leq\mathcal{B}(m,f)$. We also discuss some possible applications of thisresult.

On ascending chains of ideals in the polynomial ring