Further Study Of Geometry Of Matrices Over Finite Local Rings

The study of the geometry of matrices was initiated by L.K. Hua in the middle forties of the last century. In recent decades, it has very big development. In recent years, the conditions of the fundamental theorems of the geometry of matrices are simplified, at the same time, the research of the geometry of matrices over fields has expanded to the matrices over rings. R=Z/psZ(s≥2) is a finite local ring, also a Galois ring, it plays an important role in combinations and coding theory. The work of this paper is to discuss the geometry of matrices over Z/psZ.The content of this text is composed of four chapters. In Chapter1, we give a brief introduction of the background of the current research and the main results of the paper. In Chapter2, we discuss the properties of Z/psZ, define the p-power of the element in a Galois ring, and give some properties of the p-power. Also, the following content are discussed:inner rank, Mc-rank, determinant rank of the matrices over Z/psZ, and the theory of the matrices over Z/psZ. In Chapter3, we discuss the structure of maximal set of rank one and the affine geometry on maximal sets, and obtain an important result:if R=Z/psZ (s≥2) and J=rad(R), then in Rm×n, there is no any maximal set of rank one which is contained in Jm×n.In Chapter4, we characterize the bijective map φ:R2×n→R2×n which preserves the adjacency in both directions and the Mc-adjacency under some conditions, and characterize the additive rank one preserving bijective map as follows. Let R=Z/psZ (s≥2), m,n≥2, and φ be an additive bijective map from Rm×n to itself such that both φ and φ-1preserve the adjacency and Mc-adjacency. Then when m≠n, φ is of the form φ(X)=PXσQ for all X∈Rm×n, where P∈GLm(R), Q∈GLn(R), and a is an isomorphism from R to R’. When m=n,φ is of the form either as above or φ(X)=Pt(Xσ)Q for all X∈Rn×n, where P, Q, a have the same meaning as above.