T-dual Rickart modules

We introduce the notions of T-dual Rickart and strongly Tdual Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that every free (respectively, nitely generated free) R-module is T-dual Rickart if and only if Z² (R) is a direct summand of R and End(Z ² (R)) is a semisimple (resp. regular) ring. It is shown that, while a direct summand of a (strongly) T-dual Rickart module inherits the property, direct sums of T-dual Rickart modules do not. Moreover, when a direct sum of T-dual Rickart modules is T-dual Rickart, is included. Examples illustrating the results are presented.