@article{article_1058380, title={On a property of the ideals of the polynomial ring $R[x]$}, journal={International Electronic Journal of Algebra}, volume={31}, pages={1–12}, year={2022}, DOI={10.24330/ieja.1058380}, author={Al-maktry, Amr Ali Abdulkader}, keywords={Commutative rings, polynomial ring, null ideal, null polynomial, Henselian ring, finite local ring, dual numbers, polynomial permutation, permutation polynomial, finite permutation group}, abstract={Let <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">R </span> </span> </span> </span> <span class="MJX_Assistive_MathML">R </span> </span> be a commutative ring with unity <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">1 </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">≠ </span> </span> <span class="mjx-mn MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.378em;">0 </span> </span> </span> </span> <span class="MJX_Assistive_MathML">1≠0 </span> </span>. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">R </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">[ </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.316em;">x </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">] </span> </span> </span> </span> <span class="MJX_Assistive_MathML">R[x] </span> </span>. In particular, when <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">R </span> </span> </span> </span> <span class="MJX_Assistive_MathML">R </span> </span> is a finite local ring with principal maximal ideal <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-frak-R" style="padding-top:.253em;padding-bottom:.316em;">m </span> </span> </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">≠ </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">{ </span> </span> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.378em;">0 </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">} </span> </span> </span> </span> <span class="MJX_Assistive_MathML">m≠{0} </span> </span> of index of nilpotency <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.316em;">e </span> </span> </span> </span> <span class="MJX_Assistive_MathML">e </span> </span>, where <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">1 </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.503em;">< </span> </span> <span class="mjx-mi MJXc-space3"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.316em;">e </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.503em;">≤ </span> </span> <span class="mjx-texatom MJXc-space3"> <span class="mjx-mrow"> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">| </span> </span> </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">R </span> </span> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">/ </span> </span> </span> </span> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-frak-R" style="pa}, number={31}, publisher={Abdullah HARMANCI}