Additivity of Jordan n-tuple maps on rings

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.