An LDPC code is proposed for flash memories based on rank modulation. In contrast to previous approaches, this enables the use of long ECCs with fixed-length modulation codes. For ECC design, the rank modulation scheme is treated as part of an equivalent channel. A probabilistic model of the equivalent channel is derived and a simple high-SNR approximation is given. LDPC codes over integer rings and finite fields are designed for the approximate channel and a low-complexity symbol-flipping verification-based (SFVB) message-passing decoding algorithm is proposed to take advantage of the channel structure. Density evolution (DE) is used to calculate decoding thresholds and simulations are used to compare the low-complexity decoder with sum-product decoding.