Linear maps preserving idempotence on matrix modules over principal ideal domains (Q1359189)

Let \(R\) be a commutative principal ideal domain, \(T\colon M_n(R)\to M_m(R)\) an \(R\)-linear map which preserves idempotence. The author determines the forms of \(T\) when \(n\geq m\) and \(R\neq F_2\), and solves some of Beasley's open problems. The author proves that the set \({\mathcal L}(R)\) of all \(R\)-linear maps on \(M_n(R)\) which preserves both idempotence and non-idempotence is a proper subset of \({\mathcal F}(R)\), the set of all linear maps on \(M_n(R)\) that preserve idempotence, when the characteristic of \(R\) is 2. The methods used are different from those of Beasley's [\textit{L. B. Beasley} and \textit{N. J. Pullman}, Linear Algebra Appl. 146, 7-20 (1991; Zbl 0718.15004)].