LET R BE A RING AND M AN R-MODULE. IN THIS PAPER THE NOTION OF A T-SMALL SUBMODULEIS USED FREQUENTLY. A SUBMODULEN OF M IS T-SMALL IN M (DENOTED BY N <<T&NBSP; M), IF&NBSP; N + K Í Z2 (M) IMPLIES K Í &NBSP;Z2 (M). ALSON IS CALLED T-COCLOSED IN M, IN CASE N/K << T M/K, FOR A SUBMODULE K CONTAINED IN N, IMPLIES THAT N=K. WE SAY THAT M IS T − D11 IF EVERY T-COCLOSED SUBMODULE OF M HAS A SUPPLEMENT WHICH IS A DIRECT SUMMAND OFM. FOR AN AMPLY SUPPLEMENTED MODULE M, WE PROVE THAT M IS T − D11 IF AND ONLY IF M=Z2 (M) Å&NBSP; L WHERE Z2 (M) AND L ARE BOTH T − D11. SOME RELATIONS OF T − D11-MODULES WITH THE OTHER TYPES OF MODULES INVESTIGATED. EXAMPLES ARE PRESENTED TO SEPARATE DIFFERENT CONCEPTS.