the underlying set of the module and its operation must form a group. a module is a ring-theoretic version of a group representation: ignoring the second ring operation and the operation on pairs of module elements determines a group representation. the real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. an important special case of this is an "algebra". (the word algebra means both a subject within mathematics as well as an object studied in the subject of algebra.) an algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. this operation makes the second ring into a module over the first. often the first ring is a field.