Some Study About Affine Geometry

The affine geometry is the fundamental content of geometry, it works over therelations, propositions and the collineation between two affine geometries. The affinegeometry have many applications in mathematic and engineering. The study of theaffine geometry over a division ring have relatively maturated, in recent years, thestudy of the affine geometry over Bezout domain have many results, but the presentlyresearch about affine geometry over ring has many problem to be resolved. In recentyears, the geometry over ring is an active direction, our national scholars have mademuch contribution on it. The classical fundament theorem over a division ring is: thecollineation between two affine geometries is a semi-linear mapping. The fundamenttheorem have many application, and its generally form firstly proved by Hua Luo-Keng in 1951. The fundament theorem's simple condition and further studing is stillattention by scholars. But the result can't be directly extended to affine geometry overring. Generally speaking, the collineation between two affine geometries over rings maynot be a semi-linear mapping.Local ring is important type of rings, it have wide applications in mathematic andpractical problems. But the study of affine geometry over a local ring is very di?cult,thus the papers on its study is rare. In order to study the geometry of matrices overa local ring, this paper preliminary studies the affine geometry theory over a localring. This paper discusses some propositions on affine geometry over a local ring.What condition, the collineation between two affine geometries over local rings is asemi-linear mapping? The article preliminary answers the problem, and proves thefollowing result:Let m₁≥n₁≥2,m₂≥n₂≥2,R₁, and let R₁ and R₂ be two commutative local rings.Let V₁ and V₂ be m-dimensional free R1-module and R₂-module, respectively. Ifφisa collineation from AG(V₁) to AG(V₂) such thatφpreserves the unimodular elementsand the parallelism of lines, thenψ=φ(x) -φ(0) is a semi-linear bijective map.