A ring mapping \(\varphi\) is called a Jordan homomorphism if it is additive and satisfies \[ \varphi(xy+ yx)=\varphi(x)\varphi(y)+ \varphi(y)\varphi(x) \] for all \(x\), \(y\). Rings with Jordan structure have received much attention and Jordan operator algebras belong to the mathematical foundations of quantum mechanics. If the ring \(\mathbb{R}'\) is 2-torsion-free, then every Jordan homomorphism \(\varphi:\mathbb{R}\to \mathbb{R}'\) is a Jordan triple homomorphism, i.e. is additive and satisfies \(\varphi(xyx)= \varphi(x)\varphi(y)\varphi(x)\) for all \(x,y\in\mathbb{R}\). Without additivity, such a map is called a Jordan semi-triple map.
In this paper the authors consider the algebra \(M_n(F)\) of \(n\times n\) matrices over an arbitrary field \(F\). The main result is: Let \(n> 1\). An injective mapping \(\Phi:M_n(F)\to M_n(F)\) is a Jordan semi-triple map if and only if there exist \(\sigma\in F\), \(\sigma=\pm 1\), an invertible matrix \(T\in M_n(F)\) and an injective homomorphism \(\varphi\) of \(F\) such that either \(\Phi(A)=\sigma TA_\varphi T^{-1}\) for all \(A\) or \(\Phi(A)=\sigma T(A_\varphi)^t T^{-1}\) for all \(A\), where \(A_\varphi\) is the image of \(A\) under \(\varphi\) applied entrywise.