Additivity of Jordan n-tuple maps on rings

Abstract

Let $\mathfrak {R}$ and $\mathfrak {R}'$ be rings. We study the additivity of surjective maps $M:\mathfrak {R}\rightarrow \mathfrak {R}'$ and $M^{*}:\mathfrak {R}'\rightarrow \mathfrak {R}$ preserving a type of Jordan n-tuple product on these rings. We prove that if $\mathfrak {R}$ contains a non-trivial idempotent satisfying some conditions, then they are additive. In particular, if $\mathfrak {R}$ is a standard operator algebra, then both M and $M^{*}$ are additive.