It is shown that if S is a commutative multiplicative monoid with a zero element, the set C(S) of congruences on S is a lattice module over the multiplicative lattice L(S) of ideals of S. This gives a method of obtaining results on the set of congruences on a monoid by extending results on modules to lattice modules, and shows the relevance of examples and results on set of congruences on a monoid to the development of results on lattice modules. We illustrate these two aspects by extending specific results on modules to lattice modules, to get corresponding results on the set of congruences on a monoid, and by extending a well-known result of Drbohlav on the set of congruences on a monoid, to obtain a more general primary decomposition theorem on lattice modules than those in the literature.
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