Modules are to rings what vector spaces are to fields: elements of a given module over a ring can be added to one another and multiplied by elements of the ring. Modules arise in algebraic geometry and its applications because a geometric structure on a variety often corresponds algebraically to a module or an element of a module over the coordinate ring of the variety. Examples of geometric structures on a variety that correspond to modules in this way include subvarieties, various sets of functions, and vector fields and differential forms on a variety. In this chapter, we will introduce modules over polynomial rings (and other related rings) and explore some of their algebra, including a generalization of the theory of Gröbner bases for ideals.
KeywordsLocal Ring Polynomial Ring Free Module Monomial Ideal Division Algorithm
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