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# Inverse Limit

The inverse limit of a family of -modules is the dual notion of a direct limit and is characterized by the following mapping property. For a directed set and a family of -modules , let be an inverse system. is some -module with some homomorphisms , where for each ,

 (1)

such that if there exists some -module with homomorphisms , where for each ,

 (2)

then a unique homomorphism is induced and the above diagram commutes.

The inverse limit can be constructed as follows. For a given inverse system, , write

 (3)

Commutative Diagram, Direct Limit, Direct Sum, Direct System, Directed Set, Module, Module Homomorphism, Quotient Module

This entry contributed by Bart Snapp

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## References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Menlo Park, CA: Addison-Wesley, 1969.Matsumura, H. Commutative Ring Theory. New York: Cambridge University Press, 1986.Rotman, J. J. Advanced Modern Algebra. Upper Saddle River, NJ: Prentice Hall, 2002.

Inverse Limit

## Cite this as:

Snapp, Bart. "Inverse Limit." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseLimit.html