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Direct System

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let

sigma_(ij):M_i->M_j i<=j

be an R-module homomorphism. Call (M_i,sigma_(ij)) a direct system over I when

1. sigma_(ii)(m)=m for all m in M_i.

2. When i<=j<=k, sigma_(ik)=sigma_(jk) degreessigma_(ij).

See also

Directed Set, Inverse System, Module, Module Homomorphism

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.

Referenced on Wolfram|Alpha

Direct System

Cite this as:

Weisstein, Eric W., with contributions by Bart Snapp. "Direct System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirectSystem.html

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