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Semilocal Ring


A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal.

If K is a field, the maximal ideals of the ring K[X] of polynomials in the indeterminate X are the principal ideals

 <X-alpha>={f(X)(X-alpha)|f(X) in K[X]},

where alpha is any element of K. There is a one-to-one correspondence between these ideals and the elements of K. Hence K[X] is semilocal if and only if K is finite.

A semilocal ring always has finite Krull dimension.

The ring of integers Z is an example of a Noetherian nonsemilocal ring, since its maximal ideals are the principal ideals <p>, where p is any prime number.


See also

Local Ring, Maximal Ideal

This entry contributed by Margherita Barile

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References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.Hartley, B. and Hawkes, T. O. Rings, Modules and Linear Algebra. London, England: Chapman and Hall, 1970.Hutchins, H. H. Examples of Commutative Rings. Passaic, NJ: Polygonal Publishing House, 1981.Kunz, E. Introduction to Commutative Algebra and Algebraic Geometry. Boston, MA: Birkhäuser, 1985.Matsumura, H. Commutative Ring Theory. Cambridge, England: Cambridge University Press, 1986.Nagata, M. Local Rings. Huntington, NY: Krieger, 1975.Samuel, P. and Zariski, O. Commutative Algebra I. Princeton, NJ: Van Nostrand, 1958.Sharp, R. Y. Steps in Commutative Algebra, 2nd ed. Cambridge, England: Cambridge University Press, 2000.

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Semilocal Ring

Cite this as:

Barile, Margherita. "Semilocal Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SemilocalRing.html

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