A module over a ring , in the left-module convention, is an Abelian group together with an operation , written , satisfying

for all and (Dummit and Foote 2004, p. 337). If has a multiplicative identity 1, the additional condition defines a unital R-module. A module is similar to a vector space, but its coefficients lie in a ring rather than necessarily a field. A module over is also called an R-module.

Modules are the basic tool of homological algebra. Examples over the ring of integers include itself, the integer lattice , and the integral group ring of a group. Each uses ordinary integer multiplication as its coefficient operation.

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 for and a fixed integer form the submodule since, for all ,

and is still in .

Given two integers and , the smallest submodule of containing them is , where is their greatest common divisor.

See also

Artinian Module, Composition Series, Different, Direct Sum, Faithfully Flat Module, Field, Finitely Generated Module, Flat Module, Graded Module, Group Ring, Homological Algebra, Injective Module, Jordan-Hölder Theorem, Modular System, Module Depth, Module Discriminant, Module Length, Projective Module, Quotient Module, R-Module, Ring, Submodule, 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. "Introduction to Module Theory." Ch. 10 in Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley, pp. 337-407, 2004.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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Weisstein, Eric W. "Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Module.html

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