Inducing lattice maps by semilinear isomorphisms (Q795909)

Let R be either a finite product of rings, each of which has linearly ordered lattice of left ideals, or a semihereditary ring, or an integral domain (not assumed to be commutative), F be a free left R-module, with the basis \({\mathcal E}\), where \(Card {\mathcal E}=n\geq 3,\) S be a ring and G be a free left S-module. If \(\phi\) is an isomorphism of the lattices of submodules of F and G and \(\phi(Re)=S\) for each \(e\in {\mathcal E}\) then \(\phi\) is induced by a semilinear isomorphism [cf. \textit{W. Stephenson}, J. Lond. Math. Soc., II. Ser. 1, 177-183 (1969; Zbl 0212.380); \textit{F. Machala}, Cas. Pestovani Mat. 24, 301-306 (1974; Zbl 0309.16019); Czech. Math. J. 24(99), 26-39 (1974; Zbl 0308.50012); 25(100), 214-226 (1975; Zbl 0328.50019); 454-474 (1975; Zbl 0318.50017)].