Graph homomorphisms on rectangular matrices over division rings II. Issue 2 (1st February 2020) - Huang, Li-Ping; Zhao, Kang
ABSTRACT: Let D m × n be the set of m × n matrices over a division ring D . Two matrices A, B ∈ D m × n are adjacent if rank ( A − B ) = 1 . By the adjacency, D m × n is a connected graph. Suppose D, D ′ are division rings and m, n, m ′, n ′ ≥ 2 are integers. When | D | ≥ 4, we characterize the graph homomorphism ϕ : D n × n → D ′ m ′ × n ′ if ϕ ( 0 ) = 0 and there exists A 0 ∈ D n × n such that rank ( ϕ ( A 0 ) ) = n .