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Local Ring


A local ring is a ring R that contains a single maximal ideal. In this case, the Jacobson radical equals this maximal ideal.

One property of a local ring R is that the subset R-m is precisely the set of ring units, where m is the maximal ideal. This follows because, in a ring, any nonunit belongs to at least one maximal ideal.


See also

Jacobson Radical, Localization, Maximal Ideal, Quasilocal Ring, Regular Local Ring, Residue Field, Ring Unit, Semilocal Ring

Portions of this entry contributed by Todd Rowland

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References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

Referenced on Wolfram|Alpha

Local Ring

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Local Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LocalRing.html

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