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Module


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.

Z is a module over itself. It is closed under addition and subtraction (although it is sufficient to require closure under subtraction). Numbers of the form nalpha for n in Z and alpha a fixed integer form a submodule since, for all (n,m) in Z,

 nalpha+/-malpha=(n+/-m)alpha

and (n+/-m) is still in Z.

Given two integers a and b, the smallest module containing a and b is the module for their greatest common divisor, alpha=GCD(a,b).


See also

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, R-Module, Ring, Submodule, Verma Module, Vector Space, Zero Module Explore this topic in the MathWorld classroom

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References

Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999.Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with K-Theory in View. Cambridge, England: Cambridge University Press, 2000.Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillian, p. 390, 1996.Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.Herstein, I. N. "Modules." §1.1 in Noncommutative Rings. Washington, DC: Math. Assoc. Amer., pp. 1-8, 1968.Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951.Riesel, H. "Modules." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 239-240, 1994.

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Module

Cite this as:

Weisstein, Eric W. "Module." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Module.html

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