Understanding the dynamics of Boolean networks is central to problems such as network reduction, design, control, and reverse engineering. As Boolean network models continue to grow in size and complexity, it becomes increasingly important to decompose networks into modules in a manner that is compatible with their dynamics. In this paper, we show that endowing the space of possible dynamics with a semiring structure enables a systematic decomposition of the dynamics of any Boolean network in terms of the dynamics of its constituent modules. This algebraic framework provides a systematic way to analyze how local dynamical behaviors combine to produce global dynamics. Our results establish a concrete algebraic foundation for network modularity and introduce new mathematical tools for the study of complex Boolean networks, and opens the door to the application of algebraic methods to problems of network analysis, decomposition, and control.

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