Springer Iranian Journal of Science 2731-8095 39 4 2015 12 15 Derivations with power values on multilinear polynomials 521 525 3404 10.22099/ijsts.2015.3404 EN S. Huang School of Mathematics and Finance, Chuzhou University, China Journal Article 2014 04 05 A polynomial   <span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">1 2 </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">( , , , ) </span></span><em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">f X X X</span></span></em> is called multilinear if it is homogeneous and linear in every one of its variables. In the present paper our objective is to prove the following result: Let   <em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">R </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">be a prime </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">K</span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">-algebra over a</span></span> commutative ring   <em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">K </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">with unity and let </span></span><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">1 2 </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">( , , , ) </span></span><em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">f X X X </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">be a multilinear polynomial over </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">K</span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">. Suppose</span></span> that   <em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">d </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">is a nonzero derivation on </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">R </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">such that </span></span><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">1 2 1 2 </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">( , , , ) ( , , , ) </span></span><em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">s t</span></span></em> df x x x   <span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span> <span style="font-family: Symbol; font-size: large;" lang="ZH-TW"><span style="font-family: Symbol; font-size: large;" lang="ZH-TW"> </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">f x x x</span></span><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">for all</span></span> 1 2   <span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">, , , </span></span><em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">x x x </span></span></em><span style="font-family: Symbol; font-size: large;" lang="ZH-TW"><span style="font-family: Symbol; font-size: large;" lang="ZH-TW"></span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">R</span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">, where </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">s</span></span></em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">,</span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">t </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">are fixed positive integers. Then </span></span><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">1 2 </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">( , , , ) </span></span><em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">n </span></span><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">f X X X </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">is central-valued on</span></span> R <span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">. We also examine the case </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">R </span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">which is a semiprime </span></span><em><span style="font-family: Times New Roman; font-size: large;"><span style="font-family: Times New Roman; font-size: large;">K</span></span></em><span style="font-family: Times New Roman; font-size: xx-small;"><span style="font-family: Times New Roman; font-size: xx-small;">-algebra.</span></span> https://ijsts.shirazu.ac.ir/article_3404_95e6e817b5e96687654842c461aa75b5.pdf