Higher Derivations Satisfying Certain Identities in Rings

Let <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>n</a:mi></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>m</c:mi></c:math> be fixed positive integers. In this paper, we establish some structural properties of prime rings equipped with higher derivations. Motivated by the works of Herstein and Bell-Daif, we characterize rings with higher derivations <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:mi>D</e:mi><e:mo>=</e:mo><e:msub><e:mrow><e:mfenced open="(" close=")" separators="|"><e:mrow><e:msub><e:mrow><e:mi>d</e:mi></e:mrow><e:mrow><e:mi>i</e:mi></e:mrow></e:msub></e:mrow></e:mfenced></e:mrow><e:mrow><e:mi>i</e:mi><e:mo mathvariant="double-struck">∈</e:mo><e:mi mathvariant="double-struck">N</e:mi></e:mrow></e:msub></e:math> satisfying (i) <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M4"><l:mfenced open="[" close="]" separators="|"><l:mrow><l:msub><l:mrow><l:mi>d</l:mi></l:mrow><l:mrow><l:mi>n</l:mi></l:mrow></l:msub><l:mrow><l:mfenced open="(" close=")" separators="|"><l:mrow><l:mi>x</l:mi></l:mrow></l:mfenced></l:mrow><l:mo>,</l:mo><l:msub><l:mrow><l:mi>d</l:mi></l:mrow><l:mrow><l:mi>m</l:mi></l:mrow></l:msub><l:mrow><l:mfenced open="(" close=")" separators="|"><l:mrow><l:mi>y</l:mi></l:mrow></l:mfenced></l:mrow></l:mrow></l:mfenced><l:mo>∈</l:mo><l:mi>Z</l:mi><l:mfenced open="(" close=")" separators="|"><l:mrow><l:mi mathvariant="script">R</l:mi></l:mrow></l:mfenced></l:math> for all <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M5"><ab:mi>x</ab:mi><ab:mo>,</ab:mo><ab:mi>y</ab:mi><ab:mo>∈</ab:mo><ab:mi mathvariant="script">R</ab:mi></ab:math> and (ii) <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" id="M6"><db:msub><db:mrow><db:mi>d</db:mi></db:mrow><db:mrow><db:mi>n</db:mi></db:mrow></db:msub><db:mfenced open="(" close=")" separators="|"><db:mrow><db:mfenced open="[" close="]" separators="|"><db:mrow><db:mi>x</db:mi><db:mo>,</db:mo><db:mi>y</db:mi></db:mrow></db:mfenced></db:mrow></db:mfenced><db:mo>∈</db:mo><db:mi>Z</db:mi><db:mfenced open="(" close=")" separators="|"><db:mrow><db:mi mathvariant="script">R</db:mi></db:mrow></db:mfenced></db:math> for all <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M7"><pb:mi>x</pb:mi><pb:mo>,</pb:mo><pb:mi>y</pb:mi><pb:mo>∈</pb:mo><pb:mi mathvariant="script">R</pb:mi></pb:math> .