Extending Hua’s theorem on the geometry of matrices to Bezout domains

This paper extends Hua’s theorem on the geometry of rectangular matrices over a division ring to the case of Bezout domains. Let m,n,m′,n′ be integers ⩾2, R an R′ be two Bezout domains. Assume that φ:Rm×n→R′m′×n′ is an adjacency preserving bijective map in both directions. Further, assume that R′ is a local ring, or φ is an invertibility preserving map, or φ is an additive map. This paper obtains the algebraic formulas of φ. As applications, the ring semi-isomorphisms from Rm×n to R′m′×n′ are characterized, and the group isomorphisms from GLn(R) to GLn′(R′) are discussed.