Left Ideal Preserving Maps on Triangular Algebras

Abstract

Let A be a unital algebra over a commutative unital ring R. We say that A is a SLIP algebra if every R-linear map on A that leaves invariant every left ideal of A is a left multiplier. In this paper we study whether a triangular algebra Tri(A,M,B) is a SLIP algebra and give some necessary or sufficient conditions for a triangular algebra be a SLIP algebra, and various examples are given which illustrate limitations on extending some of the theory developed. Then our results are applied to generalized triangular matrix algebras and block upper triangular algebras. Also, some SLIP algebras other that triangular algebras are provided.