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The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex $Ho{m}_{R̂}((,R̂),M)$ is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.