• David Cox
  • John Little
  • Donal O’Shea
Part of the Graduate Texts in Mathematics book series (GTM, volume 185)


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.


Local Ring Polynomial Ring Free Module Monomial Ideal Division Algorithm 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • David Cox
    • 1
  • John Little
    • 2
  • Donal O’Shea
    • 3
  1. 1.Department of Mathematics and Computer ScienceAmherst CollegeAmherstUSA
  2. 2.Department of MathematicsCollege of the Holy CrossWorcesterUSA
  3. 3.Department of Mathematics, Statistics and Computer ScienceMount Holyoke CollegeSouth HadleyUSA

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