Linear Operators Preserving Idempotent On 2×2 Matrix Spaces

Linear Preserve Problems (LPP) is an active research area in matrix theory in the past decades. One of the reasons why the area is active is its theoretical values, another reason is it has extensive practical values. LPP is applied in many fields, such as differential equation, system control, mathematical statistics and so on.Idempotent preserver problem is one of the most important research directions in linear preserver problems, because in linear preserver problems many problems such as preserving inverse, preserving rank-additivity can always be transformed to idempotent preserver problem.This paper introduces some elementary definitions and theories in algebra, then obtain the following results:1) characterize the linear preserver of idempotence on 2×2 triangular matrix spaces over field (which is not two element field);2) characterize the additive preserver of idempotence on 2×2 triangular matrix spaces over two element field;3) characterize the additive preserver of idempotence on 2×2 matrix spaces over two element field.