Modules

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

Abstract

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.

Keywords

Local Ring Polynomial Ring Free Module Monomial Ideal Division Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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