In this paper, we introduce the notion of FI-t-lifting modules which is a proper generalization of both the concepts of t-lifting modules and FI-lifting modules. We show that a direct sum of FI-t-lifting modules is not FI-t-lifting, in general. It is also shown that if M is an FI-t-lifting module, then Z¯¯¯¯2(M) is a direct summand of M and Z¯¯¯¯2(M) is a noncosingular FI-lifting module. The last part of the paper is devoted to the study of amply supplemented FI-t-lifting modules.
