Additive Operators Preserving Idempotent Over Division Ring

The classification of linear transformations under which certain matrix properties, relations, or subsets are invariant is the object of intensive study during the last decades, because these problems usually have stronger practical effects. Characterizing linear mappings on matrix that satisfy preserving invariant, is often referred to as the research in the area of so called linear preserver problems. It plays a very important role in the algebra, because its theoretical values, and many problems in differential equation, system control, mathematical statistics and the other fields have extensive practical values. The matrix over division ring plays a very important role in many fields, Such as physics and computer graphics. But multiplicative commutative law doesn't satisfy in division ring. So it deepens difficulties over division ring .this paper quotes Liu shaowu's linear maps preserving idempotence matrix over division ring and Zhang xian's additive operators preserving idempotent matrix over fields and application. Many researchers have dicussed linear preserver problems on matrix over field. For example, there are many researches on linear maps which preserve spectrum, rank, nilpotency and idempotent and so on. This paper studies additive operators preserve idempotence over division ring. Take it as basis researching additive operators preserving idempotent matrix over division ring, additive operators preserving idempotent matrix on upper triangular matrix algebras over division, Additive operators preserving tripotent matrix over division ring, additive operators preserving tripotent matrix on upper triangular matrix over division ring, additive operators preserving {1, 2} inverse or {1} over division ring and additive operators preserving group inverse over division ring. Liu Shaowu studied linear operators preserving idempotence on matrix spaces over divison ring, Additive operators are instead of linear operators in this paper.Zhan Xian and Cao Chongguang consideded additive operators preserving idempotent matrix over fields, this paper is over division ring. But this paper has a weakness that the character of this division ring doesn't equal to 2.