Automatic additivity for injective Jordan semi-triple maps on structural matrix rings over division rings

Abstract

Let $\mathbb D$ be a division ring, and let $\mathcal{R}\subseteq M_n(\mathbb{D})$ be a structural matrix ring over $\mathbb{D}$, that is, the subring of $M_n(\mathbb{D})$ supported on the ordered pairs of a preorder on $\{1,\ldots,n\}$. We study injective Jordan semi-triple maps $\phi:\mathcal{R}\to M_n(\mathbb{D})$, namely injective maps satisfying \[ \phi(XYX)=\phi(X)\phi(Y)\phi(X), \qquad \text{for all } X,Y\in\mathcal{R}. \] Assuming that the centre of $\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. The first one is scalar: it occurs precisely when $\mathcal{R}$ has a direct ring summand isomorphic to $\mathbb{D}$ and $\mathbb{D}$ is isomorphic to neither $\mathbb{F}_3$ nor $\mathbb{F}_4$. The second one is order-theoretic: it occurs when a nonsymmetric comparable pair $i\preceq j$, $j\not\preceq i$, admits no third index $k\notin\{i,j\}$ comparable with both $i$ and $j$. If neither obstruction occurs, all injective Jordan semi-triple maps are additive. The centre-size hypothesis is sharp: for $n\ge3$, the upper-triangular ring $T_n(\mathbb{F}_2)$ has neither obstruction but nevertheless admits nonadditive injective Jordan semi-triple maps. Finally, in the additive case, we describe the maps componentwise, in terms of endomorphisms, anti-endomorphisms, and transitive multipliers.