Boolean matrix factorization (BMF) has many applications in data mining, bioinformatics, and network analysis. The goal of BMF is to decompose a given binary matrix as the Boolean product of two smaller binary matrices, revealing underlying structure in the data. When interpreting a binary matrix as the adjacency matrix of a bipartite graph, BMF is equivalent to the NP-hard biclique cover problem. By approaching this problem through the lens of commutative algebra, we utilize algebraic structures and techniques--particularly the Castelnuovo-Mumford regularity of combinatorially defined ideals--to establish new lower bounds for Boolean matrix rank.