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    Module

    Alternate name

    modul

    Basic definition

    A module is a generalization of a vector space in which the scalars form a ring rather than a field.

    Detailed definition

    A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. A module is abstractly very similar to a vector space, although in modules, coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. A module taking its coefficients in a ring R is called a module over R, or a R-module. Modules are the basic tool of homological algebra. Examples of modules include the set of integers Z, the cubic lattice in d dimensions Z^d, and the group ring of a group.

    Related terms

    Artinian module | different | direct sum | faithfully flat module | field | flat module | graded module | group ring | homological algebra | injective module | modular system | module discriminant | projective module | quotient module | ring | R-module | submodule | vector space | zero module

    Educational grade level

    graduate school level

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