Linear maps preserving idempotence on matrix modules over principal ideal domains Original Research Article

Let R be a commutative principal ideal domain, T: Mn(R) → Mm(R) an R-linear map which preserves idempotence. We determine the forms of T when n ≥ m and R ≠ F2, and solve some of Beasleyʹs open problems. As a consequence, we prove that the set image(R) of all R-linear maps on Mn(R) which preserve both idempotence and nonidempotence is a proper subset of image(R), the set of all linear maps on Mn(R) that preserve idempotence, when the characteristic of R is 2.