2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169870Let $(R,\fm)$ be a Cohen-Macaulay local ring. If $R$ has a canonical module, then there are some interesting results about duality for this situation. In this paper, we show that one can indeed obtain similar these results in the case $R$ has not a canonical module. Also, we give some characterizations of complete big Cohen-Macaulay modules of finite injective dimension and by using it some characterizations of Gorenstein modules over the $\fm$-adic completion of $R$ are obtained.20 pagesCommutative Algebra13C14, 13J10 (Primary); 13D05, 13C13 (Secondary)text