A triangular algebra has the form of an upper triangular matrix ring with elements having diagonal entries in and and upper right entries in ; and are unital algebras over a commutative ring with 1, and is an --bimodule that is faithful on each side. The center of is for all . The projection of to either or is central.
The main results in the paper assume that these projections are all of and . The first theorem considers an -linear map of satisfying for all so that and proves that on , for a derivation of and an -linear map from to with when . The second main result is the first with the assumptions that replaced with , but also requires the additional assumption that for any there is an integer so that is invertible.