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Modules

In: A Singular Introduction to Commutative Algebra

Author

Listed:
  • Gert-Martin Greuel

    (University of Kaiserslautern, Department of Mathematics)

  • Gerhard Pfister

    (University of Kaiserslautern, Department of Mathematics)

Abstract

Module theory may, perhaps, best be characterized as linear algebra over a ring. While classical commutative algebra was basically ideal theory, modules are in the centre of modern commutative algebra as a unifying approach. Formally, the notion of a module over a ring is the analogue of the notion of a vector space over a field, in the sense that a module is defined by the same axioms, except that we allow ring elements as scalars and not just field elements. Just as vector spaces appear naturally as the solution sets of systems of linear equations over a field, modules appear as solution sets of such systems over a ring. However, contrary to vector spaces, not every module has a basis and this makes linear algebra over a ring much richer than linear algebra over a field.

Suggested Citation

  • Gert-Martin Greuel & Gerhard Pfister, 2002. "Modules," Springer Books, in: A Singular Introduction to Commutative Algebra, chapter 2, pages 89-190, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04963-1_2
    DOI: 10.1007/978-3-662-04963-1_2
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