Roth’s equivalence problem in unit regular rings

Abstract

It is shown that for matrices over a unit regular ring [ A a m p ; C 0 a m p ; D ] ∼ [ A a m p ; 0 0 a m p ; D ] \left [ {\begin {array}{*{20}{c}} A & C \\ 0 & D \\ \end {array} } \right ]\sim \left [ {\begin {array}{*{20}{c}} A & 0 \\ 0 & D \\ \end {array} } \right ] if and only if there exist solutions X X and Y Y to A X − Y D = C AX - YD = C , thus providing a partial generalization to Roth’s theorem.