Integral closure

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Integral Closure of a Ring

In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. It is one of many closures in mathematics.

Let S be an integral domain with R a subring of S. An element s of S is said to be integral over R if s is a root of some monic polynomial with coefficients in R. ("Monic" means that the leading coefficient is 1, the identity element of R).

One can show that the set of all elements of S that are integral over R is a subring of S containing R; it is called the integral closure of R in S. If every element of S that is integral over R is already in R then R is said to be integrally closed in S. (So, intuitively, "integrally closed" means that R is "already big enough" to contain all the elements that are integral over R). An equivalent definition is that R is integrally closed in S iff the integral closure of R in S is equal to R (in general the integral closure is a superset of R). The terminology is justified by the fact that the integral closure of R in S is always integrally closed in S, and is in fact the smallest subring of S that contains R and is integrally closed in S.

In the special case where S is the fraction field of R, the integral closure of R in S is named simply the integral closure of R, and if R is integrally closed in S, then R is said to be integrally closed.

For example, the integers Z are integrally closed (the fraction field of Z is Q, and the elements of Q that are integral over Z are just the elements of Z (!), hence the integral closure of Z in Q is Z). The integral closure of Z in the complex numbers C is the set of all algebraic integers.

See also algebraic closure; this is a special case of integral closure when R and S are fields.

Integral Closure of an ideal

In commutative algebra there is also a concept of the integral closure of an ideal. The integral closure of an ideal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I \subset R} , usually denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline I} , is the set of all elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in R} such that there exists a monic polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n + a_{1} x^{n-1} + \ldots + a_{n-1} x^1 + a_n} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i \in I^i} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} as a root. The integral closure of an ideal is easily seen to be in the radical of an ideal.

There are alternate definitions as well.

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in \overline I} if there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in R} not contained in any minimal prime, such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c r^n \in I^n} for all sufficiently large Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in \overline I} if in the normalized blow-up of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , the pull back of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is contained in the inverse image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings.

References

  • R. Hartshorne, Algebraic Geometry, Springer-Verlag (1977)
  • M. Atiyah, I. Macdonald Introduction to commutative algebra Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
  • H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.