Linear preservers between matrix modules over connected commutative rings (Q1774981)

Let \(R\) denote a connected commutative ring with identity 1(\(R\) contains no idempotents except 0 and 1), and \(M_n(R)\) denotes the \(R \)-module of all \(n \times n\) matrices over \(R \). \(R \) is said to be idempotence-diagonalizable if every idempotent matrix over \(R \) is similar to a diagonal matrix. The authors characterize: (a) linear maps from \(M_n (R)\) to \(M_m (R)\) preserving tripotence when \(R \) is any idempotence-diagonalizable ring with the units 2 and 3 and (b) linear maps from \(M_n (R)\) to \(M_m (R)\) preserving some kinds of generalized inverses (i.e. Drazin inverses, group inverses, \{1\}-inverses, \{2\}-inverses and \{1,2\}-inverses) when \(R \) is either any idempotence-diagonalizable ring with the units 2 and 3, or any commutative principal ideal domain with at least one unit except for 1 and 2. These characterizations are obtained by means of a previous result due to the first author; the original version of this result was written in Chinese: here it is proved in a simpler way.