A module over a ring is a generalization of the concept of a vector space, where the scalars come from a ring instead of a field. This means that while vector spaces allow for addition and scalar multiplication with coefficients from a field, modules provide the same operations but with coefficients from a ring, which may not have all the nice properties of fields. Modules can be studied similarly to vector spaces and are crucial in understanding various algebraic structures, including primary ideals and computational methods.
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