全矩阵环上理想包含图的自同构群

设R是一个环,其上的理想包含图,记为Γ_I(R),是一个有向图,它以R的非平凡左理想为顶点,从R的左理想I_1到I_2有一条有向边当且仅当I_1真包含于I_2.环R上的理想关系图,记为Γ_i(R),也是一个有向图,它以R为顶点集,从R中元素A到B有一条有向边当且仅当A生成的左理想真包含于B生成的左理想.设F_q为有限域,其上n阶全矩阵环记为M_n(F_q),本文刻画了环M_n(F_q)上的理想包含图以及理想关系图的任意自同构.

Automorphisms of an Inclusion Ideal Graph over a Total Matrix Ring

The inclusion ideal graph of a ring R, written as Γ_I(R), is a directed graph which has all nontrivial left rings of R as vertex set and there is a directed edge from a vertex I₁ to a distinct vertex I₂ if and only if I₁ is properly contained in I₂. In addition, the ideal-relation graph of a ring R, written as Γ_i(R), is also a directed graph which has R as vertex set and there is a directed edge from a vertex A to a distinct vertex B if and only if the left ideal of R generated by A is properly contained in the left ideal generated by B. Let F_q be a finite field, the set of n×n matrices over F_q be denoted by M_n(F_q). In this paper, both the automorphisms of Γ_I(M_n(F_q)) and the automorphisms of Γ_i(M_n(F_q)) are characterized.