A finitely generated module over a unit ring is a left unital R-module with a finite set of generators , ..., such that every element of can be written as with . Equivalently, is a quotient module of a finite-rank free module (Bruns and Herzog 1998, Sharifi).

A Noetherian module is finitely generated, but the converse need not hold over an arbitrary ring. Over a Noetherian ring, every finitely generated module is a Noetherian module (Sharifi). Finite generation is the finiteness condition in the definition of a small Cohen-Macaulay module.

See also

Free Module, Module, Noetherian Module, Quotient Module, Small Cohen-Macaulay Module, Unit Ring, Unital R-Module

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.Sharifi, R. Abstract Algebra. Undated lecture notes, §§5.6-5.7 and 8.2. https://www.math.ucla.edu/~sharifi/notes/algebra.html.

Cite this as:

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

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