Channel-factorization aided detector (CFAD) is one of the important low-complexity detectors used in multiple input, multiple output (MIMO) receivers. Through channel factorization, this method transforms the original MIMO system into an equivalent system with a better-conditioned channel where detection is performed with a low-complexity detector; the estimate is then transferred back to the original system to obtain the final decision. Traditionally, the channel factorization is done with the lattice reduction algorithms such as the Lenstra-Lenstra-Lovász (LLL) and Seysen's algorithms with no consideration of the low-complexity detector used. In this paper, we propose a different approach: the channel factorization is designed specifically for the minimum mean-square-error (MMSE) detector that is a popular low-complexity detector in CFADs. Two new types of factorization algorithms are proposed. Type-I is LLL based, where the well-known DLLL-extended algorithm, the LLL algorithm working on the dual matrix of the extended channel matrix, is a member of this type but with a higher complexity. DLLL-extended is the best-performed factorization algorithm found in the literature, Type-II is greedy-search based where its members are differentiated with different algorithm's parameters. Type-II algorithms can provide around 0.5-1.0dB gain over Type-I algorithms and have a fixed computational complexity which is advantageous in hardware implementation.