Jordan <i>n</i> -derivations on alternative rings

Abstract In this paper, we investigate the problem of when a multiplicative Jordan n -derivation is additive in the case of alternative rings. More precisely, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℛ</m:mi> </m:math> {\mathscr{R}} be an alternative ring containing a non-trivial idempotent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {d_{1}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℛ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">ℛ</m:mi> <m:mn>11</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi mathvariant="script">ℛ</m:mi> <m:mn>12</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi mathvariant="script">ℛ</m:mi> <m:mn>21</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi mathvariant="script">ℛ</m:mi> <m:mn>22</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> {\mathscr{R}=\mathscr{R}_{11}+\mathscr{R}_{12}+\mathscr{R}_{21}+\mathscr{R}_{2% 2}} , the Pierce decomposition of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℛ</m:mi> </m:math> {\mathscr{R}} relative to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {d_{1}} satisfies: (i) if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>j</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msub> <m:mi>z</m:mi> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {(d_{i}ud_{j})z_{jk}=0} for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>z</m:mi> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msub> <m:mi mathvariant="script">ℛ</m:mi> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> {z_{jk}\in\mathscr{R}_{jk}} , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>j</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </