The Study Of Isomorphisms Of General Linear Groups Over Rings

General linear group is an important class of classical group, its isomorphic problem plays an important role in the classical group research, many scholars at home and abroad have done lots of work in this aspect. The algebraic depicting of automorphisms of general linear group over division rings has been resolved, and the research of isomorphisms of general linear group over rings also has a lot of results. In1992, the Russian mathematician Golubchik have proved an important theorem:the image of elementary subgroup of the general linear group over rings with order n can be determined by isomorphism and anti-isomorphism of full matrix rings. But his proof is too simple to understand. In2011, three Russian scholars have given a more detailed proof for Golubchik’s theorem, but there are still some leaps in their proof.Therefore, this paper gives a deep study of homomorphism and isomorphism of full matrix rings over rings, gets some new results, and then we supplement the proof of Golubchik’s theorem in order to better reading and understanding.This paper has three chapters. The first chapter is a brief introduction for topic background, research contents and main results.In Chapter2, we study the homomorphism problem for full matrix rings. Section2.1proves the following result. Let R, R’ be rings, φ:Mn(R)â†'R’ be a ring homomorphism. Assume that Then R is a subring of R’, ring Mn(R) is homomorphic to ring φ(I)R’φ(I), and rings φ(I)R’φ(I), Mn(R) are isomorphic. Using a subset of a ring to construct a matrix unit system (if it exists) is an important problem, and the associated one problem is: what conditions are met: a subset of a ring can be used to construct a matrix unit system? These two problems will be solved in Section2.1by a theorem. The results have a wide range of applications.In Chapter3, we mainly collate and supplement the hard-to-read leaps in the proof of Golubchik’s theorem. For instance, in the proof of Proposition1by three Russian mathematicians, directly set xij(r)=b(r)fij-c(r)fji.the conclusion is correct but difficult to understand, since seeing from the certification process, the expressions of b(r),c(r) may be related to the changes i,j, but why here seen as fixed? to solve these questions, this paper gives a detailed supplementary proof and then also make some simplified to the original proof. Since the Golubchik’s theorem is an important result with wide applications, the work of this chapter has some significance.